Morfismos, Vol 17, No 2, 2013 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 17 NÚMERO 2 JULIO A DICIEMBRE 2013 ISSN: 1870-6525

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VOLUMEN 17 NÚMERO 2 JULIO A DICIEMBRE 2013 ISSN: 1870-6525

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Morfismos

Special MIMS Proceedings Issue

This issue is devoted to the proceedings of the conference “Operads and Configuration Spaces” that took place at the Mediterranean Institute for the Mathematical Sciences (MIMS), Cité des Sciences, in Tunis capital city, June 18–22, 2012. This conference was part of the launch of MIMS in the region. Plenary speakers gave a series of lectures that were attended by students and young researchers from Tunisia and Algeria. The MIMS thanks Christophe Cazanave, Jeffrey Giansiracusa, Paolo Salvatore, Ines Saihi, Ismar Volić, Benjamin Walter, and all participants for making this a successful first conference. It also thanks Oscar-Randal Williams for his special contribution. Este n´ umero está dedicado a las memorias de la conferencia “Operads and Configuration Spaces” realizada en el Mediterranean Institute for Mathematical Sciences (MIMS), Cité des Sciences, en la ciudad de T´ unez, del 18 al 22 de junio de 2012. La conferencia fue parte de las actividades inaugurales del MIMS en la región. Los ponentes plenarios dieron una serie de conferencias a las que asistieron estudiantes e investigadores jóvenes de T´ unez y Argelia. El MIMS agradece a Christophe Cazanave, Jeffrey Giansiracusa, Paolo Salvatore, Ines Saihi, Ismar Volić, Benjamin Walter y todos los participantes por hacer de esta primera conferencia un éxito. También agradece a Oscar-Randal Williams por su contribución especial.

Contents - Contenido Configuration space integrals and the topology of knot and link spaces Ismar VoliÂ´c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Topological chiral homology and configuration spaces of spheres Oscar Randal-Williams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Cooperads as symmetric sequences Benjamin Walter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Moduli spaces and modular operads JeďŹ&#x20AC;rey Giansiracusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Morfismos, Vol. 17, No. 2, 2013, pp. 1â&#x20AC;&#x201C;56

Configuration space integrals and the topology of knot and link spaces Ismar VoliÂ´c

Abstract This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration space integrals, between a certain diagram complex and the deRham complex of the space of knots in dimension four or more. A generalization to spaces of links, homotopy links, and braids is also treated, as are connections to Milnor invariants, manifold calculus of functors, and the rational formality of the little balls operads.

2010 Mathematics Subject Classification: 57Q45, 57M27, 81Q30, 57R40. Keywords and phrases: configuration space integrals, Bott-Taubes integrals, knots, links, homotopy links, braids, finite type invariants, Vassiliev invariants, Milnor invariants, chord diagrams, weight systems, manifold calculus, embedding calculus, little balls operad, rational formality of configuration spaces. Contents 1 Introduction 1.1 Organization of the paper 2 Preliminaries 2.1 DiďŹ&#x20AC;erential forms and integration along the fiber 2.2 Space of long knots 1

2 5 6 6 9

The author was supported by the National Science Foundation grant DMS 1205786.

Ismar Voli´c

2.3 Finite type invariants 2.4 Configuration spaces and their compactification 3 Configuration space integrals and finite type knot invariants 3.1 Motivation: The linking number 3.2 “Self-linking” for knots 3.3 Finite type two knot invariant 3.4 Finite type k knot invariants 4 Generalization to Kn , n > 3 5 Further generalizations and applications 5.1 Spaces of links 5.2 Manifold calculus of functors and finite type invariants 5.3 Formality of the little balls operad References

10 17 20 20 22 23 30 33 38 38 43 48 52

Introduction

Configuration space integrals are fascinating objects that lie at the intersection of physics, combinatorics, topology, and geometry. Since their inception over twenty years ago, they have emerged as an important tool in the study of the topology of spaces of embeddings and in particular of spaces of knots and links. The beginnings of configuration space integrals can be traced back to Guadagnini, Martellini, and Mintchev [19] and Bar-Natan [4] whose work was inspired by Chern-Simons theory. The more topological point of view was introduced by Bott and Taubes [9]; configuration space integrals are because of this sometimes even called Bott-Taubes integrals in the literature (more on Bott and Taubes’ work can be found in Section 3.3). The point of this early work was to use configuration space integrals to construct a knot invariant in the spirit of the classical linking number of a two-component link. This invariant turned out to be of finite type (finite type invariants are reviewed in Section 2.3) and D. Thurston [51] generalized it to construct all finite type invariants. We will explain D. Thurston’s result in Section 3.4, but the idea is as follows: Given a trivalent diagram Γ (see Section 2.3), one can construct a bundle π : Conf[p, q; K3 , Rn ] −→ K3 ,

Configuration space integrals and knots

where K3 is the space of knots in R3 . Here p and q are the numbers of certain kinds of vertices in Γ and Conf[p, q; K3 , Rn ] is a pullback space constructed from an evaluation map and a projection map. The fiber of π over a knot K ∈ K3 is the compactified configuration space of p + q points in R3 , first p of which are constrained to lie on K. The edges of Γ also give a prescription for pulling back a product of volume forms on S 2 to Conf[p, q; K3 , Rn ]. The resulting form can then be integrated along the fiber, or pushed forward, to K3 . The dimensions work out so that this is a 0-form and, after adding the pushforwards over all trivalent diagrams of a certain type, this form is in fact closed, i.e. it is an invariant. Thurston then proves that this is a finite type invariant and that this procedure gives all finite type invariants. The next generalization was carried out by Cattaneo, Cotta-Ramusino, and Longoni [12]. Namely, let Kn , n > 3, be the space of knots in Rn . The main result of [12] is that there is a cochain map (1)

Dn −→ Ω∗ (Kn )

between a certain diagram complex Dn generalizing trivalent diagrams and the deRham complex of Kn . The map is given by exactly the same integration procedure as Thurston’s, except the degree of the form that is produced on Kn is no longer zero. Specializing to classical knots (where there is no longer a cochain map due to the so-called “anomalous face”; see Section 3.4) and degree zero, one recovers the work of Thurston. Cattaneo, Cotta-Ramusino, and Longoni have used the map (1) to show that spaces of knots have cohomology in arbitrarily high degrees in [13] by studying certain algebraic structures on Dn that correspond to those in the cohomology ring of Kn . Longoni also proved in [33] that some of these classes arise from non-trivalent diagrams. Even though configuration space integrals were in all of the aforementioned work constructed for ordinary closed knots, it has in recent years become clear that the variant for long knots is also useful. Because some of the applications we describe here have a slight preference for the long version, this is the space we will work with. The diﬀerence between the closed and the long version is minimal from the perspective of this paper, as explained at the beginning of Section 2.2. More recently, configuration space integrals have been generalized to (long) links, homotopy links, and braids [30, 57], and this work is summarized in Section 5.1. One nice feature of this generalization is

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that it provides the connection to Milnor invariants. This is because configuration space integrals give finite type invariants of homotopy links, and, since Milnor invariants are finite type, this immediately gives integral expressions for these classical invariants. We also describe two more surprising applications of configuration space integrals. Namely, one can use manifold calculus of functors to place finite type invariants in a more homotopy-theoretic setting as described in Section 5.2. Functor calculus also combines with the formality of the little n-discs operad to give a description of the rational homology of Kn , n > 3. Configuration space integrals play a central role here since they are at the heart of the proof of operad formality. Some details about this are provided in Section 5.3. In order to keep the focus of this paper on knot and links and keep its length to a manageable size, we will regrettably only point the reader to three other topics that are growing in promise and popularity. The first is the work of Sakai [44] and its expansion by Sakai and Watanabe [49] on long planes, namely embeddings of Rk in Rn fixed outside a compact set. These authors use configuration space integrals to produce nontrivial cohomology classes of this space with certain conditions on k and n. This work generalizes classes produced by others [14, 58] and complements recent work by Arone and Turchin [2] who show, using homotopy-theoretic methods, that the homology of Emb(Rk , Rn ) is given by a certain graph complex for n ≥ 2k + 2. Sakai has further used configuration space integrals to produce a cohomology class of K3 in degree one that is related to the Casson invariant [43] and has given a new interpretation of the Haefliger invariant for Emb(Rk , Rn ) for some k and n [44]. In an interesting bridge between two different points of view on spaces of knots, Sakai has in [44] also combined the configuration space integrals with Budney’s action of the little discs operad on Kn [10]. The other interesting development is the recent work of Koytcheff [29] who develops a homotopy-theoretic replacement of configuration space integrals. He uses the Pontryagin-Thom construction to “push forward” forms from Conf[p, q; Kn , Rn ] to Kn . The advantage of this approach is that is works over any coefficients, unlike ordinary configuration space integration, which takes values in R. A better understanding of how Koytcheff’s construction relates to the original configuration space integrals is still needed.

Configuration space integrals and knots

The third topic is the role configuration space integrals have recently played in the construction of asymptotic finite type invariants of divergence-free vector fields [25]. The approach in this work is to apply configuration space integrals to trajectories of a vector field. In this way, generalizations of some familiar asymptotic vector field invariants like asymptotic linking number, helicity, and the asymptotic signature can be derived. Lastly, some notes on the style and expositional choices we have made in this paper are in order. We will assume an informal tone, especially at times when writing down something precisely would require us to introduce cumbersome notation. To quote from a friend and coauthor Brian Munson [40], “we will frequently omit arguments which would distract us from our attempts at being lighthearted”. Whenever this is the case, a reference to the place where the details appear will be supplied. In particular, most of the proofs we present here have been worked out in detail elsewhere, and if we feel that the original source is suﬃcient, we will simply give a sketch of the proof and provide ample references for further reading. It is also worth pointing out that many open problems are stated througout and our ultimate hope is that, upon looking at this paper, the reader will be motivated to tackle some of them.

1.1

Organization of the paper

We begin by recall some of the necessary background in Section 2. We only give the basics but furnish abundant references for further reading. In particular, we review integration along the fiber in Section 2.1 and pay special attention to integration for infinite-dimensional manifolds and manifolds with corners. In Section 2.2 we define the space of long knots and state some observations about it. A review of finite type invariants is provided in Section 2.3; they will play a central role later. This section also includes a discussion of chord diagrams and trivalent diagrams. Finally in Section 2.4, we talk about configuration spaces and their Fulton-MacPherson compactification. These are the spaces over which our integration will take place. Section 3 is devoted to the construction of finite type invariants via configuration space integrals. The motivating notion of the linking number is recalled in Section 3.1, and that leads to the failed construction of the “self-linking” number in Section 3.2 and its improvement to the simplest finite type (Casson) invariant in Section 3.3. This section is at

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the heart of the paper since it gives all the necessary ideas for all of the constructions encountered from then on. Finally in Section 3.4 we construct all finite type knot invariants via configuration space integrals. Section 4 is dedicated to the description of the cochain map (1) and includes the definition of the cochain complex Dn . We also discuss how this generalizes D. Thurston’s construction that yields finite type invariants. Finally in Section 5, we give brief accounts of some other features, generalizations, and applications of configuration space integrals. More precisely, in Section 5.1, we generalize the constructions we will have seen for knots to links, homotopy links, and braids; in Section 5.2, we explore the connections between manifold calculus of functors and configuration space integrals; and in Section 5.3, we explain how configuration space integrals are used in the proof of the formality of the little n-discs operad and how this leads to information about the homology of spaces of knots.

2 2.1

Preliminaries Diﬀerential forms and integration along the fiber

The strategy we will employ in this paper is to produce differential forms on spaces of knots and links via configuration space integrals. Since introductory literature on differential forms is abundant (see for example [8]), we will not recall their definition here. We also assume the reader is familiar with integration of forms over manifolds. We will, however, recall some terminology that will be used throughout: Given a smooth oriented manifold M , one has the deRham cochain complex Ω∗ (M ) of differential forms: d

0 −→ Ω0 (M ) −→ Ω1 (M ) −→ Ω2 (M ) −→ · · · where Ωk (M ) is the space of smooth k-forms on M . The diﬀerential d is the exterior derivative. A form α ∈ Ωk (M ) is closed if dα = 0 and exact if α = dβ for some β ∈ Ωk−1 (M ). The kth deRham cohomology group of M , Hk (M ), is defined the usual way as the kernel of d modulo the image of d, i.e. the space of closed forms modulo the subspace of exact forms. All the cohomology we consider will be over R.

Configuration space integrals and knots

The wedge product, or exterior product, of differential forms gives Ω∗ (M ) the structure of an algebra, called the deRham, or exterior algebra of M . According to the deRham Theorem, deRham cohomology is isomorphic to the ordinary singular cohomology. In particular, H0 (M ) is the space of functionals on connected components of M , i.e. the space of invariants of M . The bulk of this paper is concerned with invariants of knots and links. The complex Ω∗ (M ) can be defined for manifolds with boundary by simply restricting the form to the boundary. Locally, we take restrictions of forms on open subsets of Rk to Rk−1 × R+ . Further, one can define differential forms on manifolds with corners (an n-dimensional manifold with corners is locally modeled on Rk+ × Rn−k , 0 ≤ k ≤ n; see [23] for a nice introduction to these spaces) in exactly the same fashion by restricting forms to the boundary components, or strata of M . The complex Ω∗ (M ) can also be defined for infinite-dimensional manifolds such as the spaces of knots and links we will consider here. One usually considers the forms on the vector space on which M is locally modeled and then patches them together into forms on all of M . When M satisfies conditions such as paracompactness, this “patchedtogether” complex again computes the ordinary cohomology of M . Another way to think about forms on an infinite-dimensional manifold M is via the diffeological point of view which considers forms on open sets mapping into M . For more details, see [30, Section 2.2] which gives further references. Given a smooth fiber bundle π : E → B whose fibers are compact oriented k-dimensional manifolds, there is a map (2)

π∗ : Ωn (E) −→ Ωn−k (B)

called the pushforward or integration along the fiber. The idea is to define the form on B pointwise by integrating over each fiber of π. Namely, since π is a bundle, each point b ∈ B has a k-dimensional neighborhood Ub such that π −1 (b) ∼ = B ×Ub . Then a form α ∈ Ωn (E) can be restricted to this fiber, and the coordinates on Ub can be “integrated away”. The result is a form on B whose dimension is that of the original form but reduced by the dimension of the fiber. The idea is to then patch these values together into a form on B. Thus the map (2) can be described

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by (3)

α !−→

b !−→

π −1 (b)

α .

In terms of evaluation on cochains, π∗ α can be thought of as an (n − k)-form on B who value on a k-chain X is " " π∗ α = α. X

π −1 (X)

For introductions to the pushforward, see [8, 21]. Remark 2.1.1. The assumption that the fibers of π be compact can be dropped, but then forms with compact support should be used to guarantee the convergence of the integral. To check if π∗ α is a closed form, it suﬃces to integrate dα, i.e. we have [8, Proposition 6.14.1] dπ∗ α = π∗ dα. The situation changes when the fiber is a manifold with boundary or with corners, as will be the case for us. Then by Stokes’ Theorem, dπ∗ α has another term. Namely, we have (4)

dπ∗ α = π∗ dα + (∂π)∗ α

where (∂π)∗ α is the integral of the restriction of α to the codimension one boundary of the fiber. This can be seen by an argument similar to the proof of the ordinary Stokes’ Theorem. For Stokes’ Theorem for manifolds with corners, see, for example [31, Chapter 10]. In the situations we will encounter here, α will be a closed form, in which case dα = 0. Thus the first term in the above formula vanishes, so that we get (5)

dπ∗ α = (∂π)∗ α.

Our setup will combine various situations described above – we will have a smooth bundle π : E → B of infinite-dimensional spaces with fibers that are finite-dimensional compact manifolds with corners.

Configuration space integrals and knots

2.2

Space of long knots

We will be working with long knots and links (links will be defined in Section 5.1), which are easier to work with than ordinary closed knots and links in many situations. For example, long knots are Hspaces via the operation of stacking, or concatenation, which gives their (co)homology groups more structure. Also, the applications of manifold calculus of functors to these spaces (Section 5.2) prefer the long model. Working with long knots is not much diﬀerent than working with ordinary closed ones since the theory of long knots in Rn is the same as the theory of based knots in S n . From the point of view of configuration space integrals, the only diﬀerence is that, for the long version, we will have to consider certain faces at infinity (see Remark 2.4.4). Before we define long knots, we remind the reader that a smooth map f : M → N between smooth manifolds M and N is • an immersion if the derivative of f is everywhere injective; • an embedding if it is an immersion and a homeomorphism onto its image. Now let Mapc (R, Rn ) be the space of smooth maps of R to Rn , n ≥ 3, which outside the standard interval I (or any compact set, really) agrees with the map R −→ Rn

t $−→ (t, 0, 0, ..., 0).

2]).

Give Mapc (R, Rn ) the C ∞ topology (see, for example, [22, Chapter

Definition 2.2.1. Define the space of long (or string) knots Kn ⊂ Mapc (R, Rn ) as the subset of maps K ∈ Mapc (R, Rn ) that are embeddings, endowed with the subspace topology. A related space that we will occassionally have use for is the space of long immersions Immc (R, Rn ) defined the same way as the space of long knots except its points are immersions. Note that Kn is a subspace of Immc (R, Rn ). A homotopy in Kn (or any other space of embeddings) is called an isotopy. A homotopy in Immc (R, Rn ) is a regular homotopy.

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K ∈ Kn Figure 1: An example of a knot in Rn .

An example of a long knot is given in Figure 1. Note that we have confused the map K with its image in Rn . We will do this routinely and it will not cause any issues. From now on, the adjective “long” will be dropped; should we need to talk about closed knots, we will say this explicitly. The space Kn is a smooth inifinite-dimensional paracompact manifold [30, Section 2.2], which means that, as explained in Section 2.1, we can consider diﬀerential forms on it and study their deRham cohomology. Classical knot theory (n = 3) is mainly concerned with computing • H0 (K3 ), which is generated (over R; recall that in this paper, the coeﬃcient ring is always R) by knot types, i.e. by isotopy classes of knots; and • H0 (K3 ), the set of knot invariants, namely locally constant (Rvalued) functions on K3 . These are therefore precisely the functions that take the same value on isotopic knots. The question of computation of knot invariants will be of particular interest to us (see Section 3). However, higher (co)homology of Kn is also interesting, even for n > 3. Of course, in this case there is no knotting or linking (by a simple general position argument), so H0 and H0 are trivial, but one can then ask about H>0 and H>0 . It turns out that our configuration space integrals contain much information about cohomology in various degrees and for all n > 3 (see Section 4).

2.3

Finite type invariants

An interesting set of knot invariants that our configuration space integrals will produce are finite type, or Vassiliev invariants. These invari-

Configuration space integrals and knots

ants are conjectured to separate knots, i.e. to form a complete set of invariants. To explain, there is no known invariant or a set of invariants (that is reasonably computable) with the following property: Given two non-isotopic knots, there is an invariant in this set that takes on different values on these two knots. The conjecture that finite type invariants form such a set of invariants has been open for some twenty years. There is some evidence that this might be true since finite type invariants do separate homotopy links [20] and braids [6, 24] (see Section 5.1 for the definitions of these spaces). Since finite type invariants will feature prominently in Section 3, we give a brief overview here. In addition to the separation conjecture, these invariants have received much attention because of their connection to physics (they arise from Chern-Simons Theory), Lie algebras, three-manifold topology, etc. The literature on finite type invariants is abundant, but a good start is [5, 15]. Suppose V is a knot invariant, so V ∈ H0 (K3 ). Consider the space of singular links, which is the subspace of Immc (R, Rn ) consisting of immersions that are embeddings except for a finite number of doublepoint self-intersections at which the two derivatives (coming from traveling through the singularity along two different pieces of the knot) are linearly independent. Each singularity can be locally “resolved” in two natural ways (up to isotopy), with one strand pushed off the other in one of two directions. A k-singular knot (a knot with k self-intersections) can thus be resolved into 2k ordinary embedded knots. We can then define V on singular knots as the sum of the values of V on those resolutions, with signs as prescribed in Figure 2. The expression given in that picture is called the Vassiliev skein relation and it depicts the situation locally around a singularity. The rest of the knot is the same for all three pictures. The knot should be oriented (by, say, the natural orientation of R); otherwise the two resolutions can be rotated into one another. Definition 2.3.1. A knot invariant V is finite type k (or Vassiliev of type k) if it vanishes on singular knots with k + 1 self-intersections. Example 2.3.2. There is only one (up to constant multiple) type 0 invariant, since such an invariant takes the same value on two knots

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−V

Figure 2: Vassiliev skein relation.

that only differ by a crossing change. Since all knots are related by crossing changes, this invariant must take the same value on all knots. It is not hard to see that there is also only one type 1 invariant. Example 2.3.3. The coefficients of the Conway, Jones, HOMFLY, and Kauffman polynomials are all finite type invariants [4, 7]. Let Vk be the real vector space generated by all finite type k invariants and let # V= Vk . k

This space is filtered; it is immediate from the definitions that Vk ⊂ Vk+1 . One of the most interesting features of finite type invariants is that a value of a type k invariant V on a k-singular knot only depends on the placement of the singularities and not on the immersion itself. This is due to a simple observation that, if a crossing of a k-singular knot is changed, the diﬀerence of the evaluation of V on the two knots (before and after the switch) is the value of V on a (k + 1)-singular knot by the Vassiliev skein relation. But V is type k so the latter value is zero, and hence V “does not see” the crossing change. Since one can get from any singular knot to any other singular knot that has the singularities in the same place (“same place” in the sense that for both knots, there are 2k points on R that are partitioned in pairs the same way; these pairs will make up the k singularities upon the immersion of R in R3 ), V in fact takes the same value on all k singular knots with the same singularity pattern. The notion of what it means for singularities to be in the “same place” warrants more explanation and leads to the beautiful and rich connections between finite type invariants and the combinatorics of chord diagrams as follows.

Configuration space integrals and knots

Definition 2.3.4. A chord diagram of degree k is a connected graph (one should think of a 1-dimensional cell complex) consisting of an oriented line segment and 2k labeled vertices marked on it (considered up to orientation-preserving diﬀeomorphism of the segment). The graph also contains k oriented chords pairing oﬀ the vertices (so each vertex is connected to exactly one other vertex by a chord). We will refer to labels and orientations as decorations and, when there is no danger of confusion, we will sometimes draw diagrams without them. Examples of chord diagrams are given in Figure 3. The reader might wish for a more proper combinatorial (rather than descriptive) definition of a chord diagram, and such a definition can be found in [30, Section 3.1] (where the definition is for the case of trivalent diagrams which we will encounter below, but it specializes to chord diagrams as the latter are a special case of the former).

Figure 3: Examples of chord diagrams. The left one is of degree 2 and the right one is of degree 3. We will always assume the segment is oriented from left to right. Definition 2.3.5. Define CDk to be the real vector space generated by chord diagrams of degree k modulo the relations 1. If Γ contains more than one chord connecting two vertices, then Γ = 0; 2. If a diagram Γ diﬀers from Γ′ by a relabeling of vertices or orientations of chords, then Γ − (−1)σ Γ′ = 0 where σ is the sum of the order of the permutation of the labels and the number of chords with diﬀerent orientation; 3. If Γ contains a chord connecting two consecutive vertices, then Γ = 0 (this is the one-term, or 1T relation);

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− a

−

= a

k l

Figure 4: The four-term (4T) relation. Chord orientations have been omitted, but they should be the same for all four pictures.

4. If four diagrams diﬀer only in portions as pictured in Figure 4, then their sum is 0 (this is the four-term, or 4T relation). We can now also define the graded vector space ! CDk . CD = k

Let CW k = Hom(CDk , R), the dual of CDk . This is called the space of weight systems of degree k, and is by definition the space of functionals on chord diagrams that vanish on 1T and 4T relations. We can now define a function f : Vk −→ CW k " W : CDk V −→ Γ

−→ R #−→ V (KΓ )

where KΓ is any k-singular knot with singularities as prescribed by Γ. By this we mean that there are 2k points on R labeled the same way as in Γ, and if x and y are points for which there exists a chord in Γ, then KΓ (x) = KΓ (y). The map f is well-defined because of the observation that type k invariant does not depend on the immersion when evaluated on a k-singular knot. The reason that the image of f is indeed in CW k , i.e. the reason that the function W we defined above vanishes on the 1T and 4T relations is not hard to see (in fact, 1T and 4T relations are part of the definition of CDk precisely because W vanishes on them): 1T relation corresponds to the singular knot essentially having a loop at the singularity; resolving those in two ways results in two isotopic knots on which V has to take the same value (since it is an invariant). Thus the diﬀerence of those values

Configuration space integrals and knots

is zero, but by the skein relation, V is then zero on a knot containing such a singularity. The vanishing on the 4T relation arises from the fact that passing a strand around a singularity of a (k − 1)-singular knot introduces four k-singular knots, and the 4T relation reflects the fact that at the end one gets back to the same (k − 1)-singular knot. It is also immediate from the definitions that the kernel of f is precisely type k − 1 invariants, so that we have an injection (which we will denote by the same letter f ) (6)

f : Vk /Vk−1 !→ CW k .

The following theorem is usually referred to as the Fundamental Theorem of Finite Type Invariants, and is due to Kontsevich [27]. Theorem 2.3.6. The map f from equation (6) is an isomorphism. Kontsevich proves this remarkable theorem by constructing the inverse to f , a map defined by integration that is now known as the Kontsevich Integral. There are now several proof of this theorem, and the one relevant to us gives the inverse of f in terms of configuration space integrals. See Remark 3.4.2 for details. Lastly we describe an alternative space of diagrams that will be better suited for our purposes. Definition 2.3.7. A trivalent diagram of degree k is a connected graph consisting of an oriented line segment (considered up to orientationpreserving diﬀeomorphism) and 2k labeled vertices of two types: segment vertices, lying on the segment, and free vertices, lying oﬀ the segment. The graph also contains some number of oriented chords connecting segment vertices and some number of oriented edges connecting two free vertices or a free vertex and a segment vertex. Each vertex is trivalent, with the segment adding two to the count of the valence of a segment vertex. Note that chord diagrams as described in Definition 2.3.4 are also trivalent diagrams. Examples of trivalent diagrams that are not chord diagrams are given in Figure 5. As mentioned before, a more combinatorial definition of trivalent diagrams is given in [30, Section 3.1]. Analogously to Definition 2.3.5, we have

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Figure 5: Examples of trivalent diagrams (that are not chord diagrams). The left one is of degree 4 and the right one is of degree 3.

Definition 2.3.8. Define T Dk to be the real vector space generated by trivalent diagrams of degree k modulo the relations 1. If Γ contains more than one edge connecting two vertices, then Γ = 0; 2. If a diagram Γ differs from Γ′ by a relabeling of vertices or orientations of chords or edges, then Γ − (−1)σ Γ′ = 0 where σ is the sum of the order of the permutation of the labels and the number of chords and edges with different orientation; 3. If Γ contains a chord connecting two consecutive segment vertices, then Γ = 0 (this is same 1T relation from before); 4. If three diagrams differ only in portions as pictured in Figure 6, then their sum is 0 (these are called the STU and IHX relation, respectively.). Finally let TD=

! k

T Dk .

Remark 2.3.9. The IHX relations actually follows from the STU relation [5, Figure 9], but it is important enough that it is usually left in the definition of the space of trivalent diagrams (it gives a connection between finite type invariants and Lie algebras). Theorem 2.3.10 ([5], Theorem 6). There is an isomorphism CDk ∼ = T Dk .

Configuration space integrals and knots

−

i = j

−

− j

i I

Figure 6: The STU and the IHX relations.

The isomorphism is given by using the STU relation repeatedly to remove all free vertices and turn a trivalent diagram into a chord diagram. The relationship between the STU and the 4T relation is that the latter is the diﬀerence of two STU relations applied to two diﬀerent edges of the “tripod” diagram in Figure 11. One can now again define the space of weight systems of degree k for trivalent diagrams as the space of functionals on T Dk that vanish on the 1T, STU, and IHX relations. We will denote these by T W k . From Theorem 2.3.10, we then have (7)

2.4

CW k ∼ = T Wk.

Configuration spaces and their compactifications

At the heart of the construction of our invariants and other cohomology classes of spaces of knots are configuration spaces and their compactifications. Definition 2.4.1. The configuration space of p points in a manifold M is the space Conf(p, M ) = {(x1 , x2 , ..., xp ) ∈ M p : xi ̸= xj for i ̸= j}

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Thus Conf(p, M ) is M p with all the diagonals, i.e. the fat diagonal, taken out. We take Conf(0, M ) to be a point. Since configuration spaces of p points on R or on S 1 have p! components, we take Conf(p, R) and Conf(p, S 1 ) to mean the component where the points x1 , ..., xp appear in linear order (i.e. x1 < x2 < · · · < xp on R or the points are encountered in that order as the circle is traversed counterclockwise starting with x1 ). Example 2.4.2. It is not hard to see that Conf(1, Rn ) = Rn and Conf(2, Rn ) ≃ S n−1 (where by “≃” we mean homotopy equivalence). The latter equivalence is given by the Gauss map which gives the direction between the two points: (8)

φ : Conf(2, Rn ) −→ S n−1 x1 − x2 . (x1 , x2 ) $−→ |x1 − x2 |

We will want to integrate over Conf(p, Rn ), but this is an open manifold and our integral hence may not converge. The “correct” compactification to take, in the sense that it has the same homotopy type as Conf(p, Rn ) is due to [3, 16], and is defined as follows. Recall that that, given a submanifold N of a manifold M , the blowup of M along N , Bl(M, N ), is obtained by replacing N by the unit normal bundle of N in M . Definition 2.4.3. Define the Fulton-MacPherson compactification of Conf(k, M ), denoted by Conf[k, M ], to be the closure of the image of the inclusion ! Bl(M S , ∆S ), Conf(p, M ) −→ M p × S⊂{1,...,p}, |S|≥2

where M S is the product of the copies of M indexed by S and ∆S is the (thin) diagonal in M S , i.e. the set of points (x, ..., x) in M S . Here are some properties of Conf[k, M ] (details and proofs can be found in [45]): • Conf[k, M ] is homotopy equivalent to Conf(k, M ); • Conf[k, M ] is a manifold with corners, compact when M is compact;

Configuration space integrals and knots

• The boundary of Conf[k, M ] is characterized by points colliding with directions and relative rates of collisions kept track of. In other words, three points colliding at the same time gives a different point in the boundary than two colliding, then the third joining them; • The stratification of the boundary is given by stages of collisions of points, so if three points collide at the same time, the resulting limiting configuration lies in a codimension one stratum. If two come together and then the third joins them, this gives a point in a codimension two stratum since the collision happened in two stages. In general, a k-stage collision gives a point in a codimension k stratum. • In particular, codimension one boundary of Conf[k, M ] is given by points colliding at the same time. This will be important in Section 4, since integration along codimension one boundary is required for Stokes’ Theorem. • Collisions can be eﬃciently encoded by diﬀerent parenthesizations, e.g. the situations described two items ago are parenthesized as (x1 x2 x3 ) and ((x1 x2 )x3 ). Since parenthesizations are related to trees, the stratification of Conf[k, M ] can thus also be encoded by trees and this leads to various connections to the theory of operads (we will not need this here); • Conf[k, R] is the associahedron, a classical object from homotopy theory; Additional discussion of the stratification of Conf[p, M ] can be found in [30, Section 4.1]. Remark 2.4.4. Since we will consider configurations on long knots, and these live in Rn , we will think of Conf[p, Rn ] as the subspace of Conf[p+ 1, S n ] where the last point is fixed at the north pole. Consequently, we will have to consider additional strata given by points escaping to infinity, which corresponds to points colliding with the north pole. Remark 2.4.5. All the properties of the compactifications we mentioned hold equally well in the case where some, but not all, diagonals are blown up. One can think of constructing the compactification by blowing up the diagonals one at a time, and the order in which we blow

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up does not matter. Upon each blowup, one ends up with a manifold with corners. This “partial blowup” will be needed in the proof of Proposition 3.3.1 (see also Remark 3.3.2).

Configuration space integrals and finite type knot invariants

3.1

Motivation: The linking number

Let Mapc (R ! R, R3 ) be the space of smooth maps which are fixed outside some compact set as in the setup leading to Definition 2.2.1 (see Definition 5.1.1 for details) and define the space of long (or string) links with two components, L32 , as the subspace of Mapc (R ! R, R3 ) given by embeddings. Now recall the definition of the configuration space from Definition 2.4.1 and the Gauss map φ from Example 2.4.2. Consider the map φ

Φ : R × R × L32 −→ Conf(2, R3 ) −→ S 2 (x1 , x2 , L = (K1 , K2 )) %−→ (K1 (x1 ), K2 (x2 )) %−→

K2 (x2 ) − K1 (x1 ) |K2 (x2 ) − K1 (x1 )|

So ev is the evaluation map which picks oﬀ two points in R3 , one on each of the strands in the image of a link L ∈ L32 , and φ records the direction between them. The picture of Φ is given in Figure 7. K2 (x2 ) Φ K1 K2

K1 (x1 )

Figure 7: The setup for the computation of the linking number. Also consider the projection map π : R × R × L32 −→ L32 which is a trivial bundle and we can thus perform integration along the fiber on it as described in Section 2.1.

Configuration space integrals and knots

Putting these maps together, we have a diagram R × R × L32

! S2

L32 which, on the complex of deRham cochains (diﬀerential forms), gives a diagram Ω∗ (R × R × L32 ) # "

Φ∗

Ω∗ (S 2 )

π∗

Ω∗−2 (L32 ) Here Φ∗ is the usual pullback and π∗ is the integration along the fiber. We will now produce a form on L32 by starting with a particular form on S 2 . So let symS 2 ∈ Ω2 (S 2 ) be the unit symetric volume form on S 2 , i.e. x dydz − y dxdz + z dxdy symS 2 = . 4π(x2 + y 2 + z 2 )3/2 This is the form that integrates to 1 over S 2 and is rotation-invariant. Let α = Φ∗ (symS 2 ). Definition 3.1.1. The linking number of the link L = (K1 , K2 ) is ! α. Lk(K1 , K2 ) = π∗ α = R×R

The expression in this definition is the famous Gauss integral. Since both the form symS 2 and the fiber R × R are two dimensional, the resulting form is 0-dimensional, i.e. it is a function that assigns a number to each two-component link. The first remarkable thing is that this form is actually closed, so that the linking number is an element of H0 (L32 ), an invariant. The second remarkable thing is that the linking number is actually an integer because it essentially computes the degree of the Gauss map. Another way to think about this is that the linking number counts the number of times one strand of L goes over the other in a projection of the link, with signs. Remark 3.1.2. The reader should not be bothered by the fact that the domain of integration is not compact. As will be shown in the proof of Proposition 3.3.1, the integral along faces at infinity vanishes.

3.2

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“Self-linking” for knots

One could now try to adapt the procedure that produced the linking number to a single knot in hope of producing some kind of a “selflinking” knot invariant. The picture describing this is given in Figure 8. K(x1 ) Φ K

K(x2 )

Figure 8: The setup for the attempted computation of a self-linking number. Since the domain of the knot is R, we now take Conf(2, R) rather than R × R. Thus the corresponding diagram is Conf(2, R) × K3

! S2

K3 The first issue is that the integration over the fiber Conf(2, R) may not converge since this space is open. One potential fix is to use the Fulton-MacPherson compactification from Definition 2.4.3 and replace Conf(2, R) by Conf[2, R]. This indeed takes care of the convergence issue, and we now have a 0-form whose value on a knot K ∈ K3 is # " ! ! K(x1 ) − K(x2 ) ∗ (9) A(K) = symS 2 = φ∗ symS 2 |K(x1 ) − K(x2 )| Conf[2,R]

Conf[2,R]

(The reason we are denoting this by A(K) is that it will have something to do with the so-caled “anomalous correction” in Section 3.4.) Checking if this form is closed comes down to checking that (5) is satisfied (symS 2 is closed so the first term of (4) goes away). In other words, we need to check that the restriction of the above integral to the face where two points on R collide vanishes. However, there is no reason for this to be true.

Configuration space integrals and knots

More precisely, in the stratum where points x1 and x2 in Conf[2, R] collide, which we denote by ∂x1 =x2 Conf[2, R], the Gauss map becomes the normalized derivative K ′ (x1 ) . |K ′ (x1 )|

(10)

Pulling back symS 2 via this map to ∂x1 =x2 Conf[2, R] × K3 and integrating over ∂x1 =x2 Conf[2, R], which is now 1-dimensional, produces a 1-form which is the boundary of π∗ α: dπ∗ α(K) =

∂x1 =x2 Conf[2,R]

K ′ (x1 ) |K ′ (x1 )|

#∗

symS 2 .

Since this integral is not necessarily zero, π∗ α fails to be invariant. One resolution to this problem is to look for another term which will cancel the contribution of dπ∗ α. As it turns out, that correction is given by the framing number [38]. The strategy for much of what is to come is precisely what we have just seen: We will set up generalizations of this “self-linking” integration and then correct them with other terms if they fail to give an invariant. Remark 3.2.1. The integral π∗ α described here is related to the familiar writhe. One way to say why our integral fails to be an invariant is that the writhe is not an invariant – it fails on the Type I Reidemeister move.

3.3

A finite type two knot invariant

It turns out that the next interesting case generalizing the “self-linking” integral from Section 3.2 is that of four points and two directions as pictured in Figure 9. The two maps Φ now have subscripts to indicate which points are being paired oﬀ (the variant where the two maps are Φ12 and Φ34 does not yield anything interesting essentially because of the 1T relation from Definition 2.3.5). Diagrammatically, the setup can be encoded by the chord diagram . This diagram tells us how many points we are evaluating a knot on and which points are being paired oﬀ. This is all

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K(x3 ) Φ13 K

K(x2 ) Φ24 K(x4 )

K(x1 )

Figure 9: Toward a generalization of the self-linking number.

the information that is needed to set up the maps Conf[4, R] × K3

Φ=Φ13 ×Φ24

! S2 × S2

K3 Let sym2S 2 be the product of two volume forms on S 2 × S 2 and let α = Φ∗ (sym2S 2 ). Since α and Conf[4, R], the fiber of π, are both 4-dimensional, integration along the fiber of π thus yields a 0-form π∗ α which we will denote by (IK3 ) ∈ Ω0 (K3 ).

By construction, the value of this form on a knot K ∈ K3 is ! (K) = α (IK3 ) π −1 (K)=Conf[4,R]

The question now is if this form is an element of H0 (K3 ). This amounts to checking if it is closed. Since symS 2 is closed, this question reduces by (5) to checking whether the restriction of π∗ α to the codimension one boundary vanishes: ! ? (K) = α|∂ = 0 d(IK3 ) ∂ Conf[4,R]

There is one such boundary integral for each stratum of ∂ Conf[4, R], and we want the sum of these integrals to vanish. We will consider various types of faces:

Configuration space integrals and knots

• prinicipal faces, where exactly two points collide; • hidden faces, where more than two points collide; • the anomalous face, the hidden face where all points collide (this face will be important later); • faces at infinity, where one or more points escape to infinity. The principal and hidden faces of Conf[4, R3 ] can be encoded by by diagrams in Figure 10, which are obtained from diagram contracting segments between points (this mimics collisions). The loop in the three bottom diagrams corresponds to the derivative map since this is exactly the setup that leads to equation (10). In other words, for each loop, the map with which we pull back the volume form is the derivative.

1=2

1=2=3

2=3

2=3=4

3=4

1=2=3=4

Figure 10: Diagrams encoding collisions of points.

Proposition 3.3.1. The restrictions of (IK3 ) faces at infinity vanish.

to hidden faces and

Proof. Since there are no maps keeping track of directions between various pairs of points on the knot, for example between K(x1 ) and K(x2 ) or K(x2 ) and K(x3 ), the blowup along those diagonals did not need to be performed. As a result, the stratum where K(x1 ) = K(x2 ) = K(x3 ) is in fact codimension two (three points moving on a one-dimensional manifold became one point) and does not contribute to the integral. The same is true for the stratum where K(x2 ) = K(x3 ) = K(x4 ). This is an instance of what is sometimes refered to as the disconnected stratum. The details of why integrals over such a stratum vanish are given in [56, Proposition 4.1]. This argument for vanishing does not work for principal faces since, when two points collide, this gives a codimension one face regardless of whether that diagonal was blown up or not.

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For the anomalous face, we have the integral # " ′ ! K ′ (x1 ) ∗ K (x1 ) × sym2S 2 . |K ′ (x1 )| |K ′ (x1 )| ∂A Conf[4,R]

The map ΦA (the extension of Φ to the anomalous face) then factors as ΦA

∂A Conf[4, R] × K3 "

! S2 × S2 #

The pullback thus factors through S 2 and, since we are pulling back a 4-form to a 2-dimensional manifold, the form must be zero. Now suppose a point, say x1 , goes to infinity. The map Φ13 is constant on this stratum so that the extension of Φ to this stratum again factors through S 2 . If more than one point goes to infinity, the factorization is through a point since both maps are constant. Remark 3.3.2. It is somewhat strange that one of the vanishing arguments in the above proof required us to essentially go back and change the space we integrate over. Fortunately, in light of Remark 2.4.5, this is not such a big imposition. The reason this happened is that, when constructing the space Conf[4, R], we only paid attention to how many points there were on the diagram and not how they were paired off. The version of the construction where all the diagram information is taken into account from the beginning is necessary for constructing integrals for homotopy links as will be described in Section 5.1. There is however no reason for the integrals corresponding to the principal faces (top three diagrams in Figure 10) to vanish. One way around this is to look for another space to integrate over which has the same three faces and subtract the integrals. This difference will then be an invariant. To find this space, we again turn to diagrams. The diagram that fits what we need is given in Figure 11 since, when we contract edges to get 4 = 1, 4 = 2, and 4 = 3, the result is same three relevant pictures as before (up to relabeling and orientation of edges). The picture suggests that we want a space of four configuration points in R3 , three of which lie on a knot, and we want to keep track of three directions between the points on the knot and the one off the knot

Configuration space integrals and knots

Figure 11: Diagram whose edge contractions give top three diagrams of Figure 10.

K(x2 )

K(x1 )

Φ14 Φ24

Φ34 K(x3 )

Figure 12: The situation schematically given by the diagram from Figure 11.

(since the diagram has those three edges). In other words, we want the situation from Figure 12. To make this precise, consider the pullback space (11)

Conf[3, 1; K3 , R3 ]

! Conf[4, R3 ]

proj

Conf[3, R] × K3

eval

! Conf[3, R3 ]

where eval is the evaluation map and proj the projection onto the first there points of a configuration. There is now an evident map π ′ : Conf[3, 1; K3 , R3 ] −→ K3 whose fiber over K ∈ K3 is precisely the configuration space of four points, three of which are constrained to lie on K. Proposition 3.3.3 ([9]). The map π ′ is a smooth bundle whose fiber is a finite-dimensional smooth manifold with corners. This allows us to perform integration along the fiber of π ′ . So let Φ = Φ14 × Φ24 × Φ34 : Conf[3, 1; K3 , R3 ] −→ (S 2 )3

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be the map giving the three directions as in Figure 12. The relevant maps are thus Conf[3, 1; K, R3 ]

! (S 2 )3

π′

α′

= Φ∗ (sym3S 2 ). This form As before, let fiber Conf[3, 1; K, R3 ] over K. Notice that

can be integrated along the both the form and the fiber are now 6-dimensional, so integration gives a form (IK3 )

∈ Ω0 (K3 ).

The value of this form on K ∈ K3 is thus ! (IK3 ) (K) =

α′

(π ′ )−1 (K)=Conf[3,1;,K,R3 ]

We now have the analog of Proposition 3.3.1: Proposition 3.3.4. The restrictions of (IK3 ) to the hidden faces and faces at infinity vanish. The same is true for the two principal faces given by the collisions of two points on the knot. Proof. The same arguments as in Proposition 3.3.1 work here, although an alternative is possible: For any of the hidden faces or the two principal faces from the statement of the proposition, at least two of the maps will be the same. For example, if K(x1 ) = K(x2 ), then Φ14 = Φ24 and Φ hence factors through a space of strictly lower dimension than the dimension of the form. A little more care is needed for faces at infinity. If some, but not all points go to infinity (this includes x4 going to infinity in any direction), the same argument as in Proposition 3.3.1 holds. If all points go to infinity, then it can be argued that, yet again, the map Φ factors through a space of lower dimension, but this has to be done a little more carefully; see proof of [30, Proposition 4.31] for details. We then have Theorem 3.3.5. The map K −→ R " K $−→ (IK3 )

(K) − (IK3 )

(K)

Configuration space integrals and knots

is a knot invariant, i.e. an element of H0 (K3 ). Further, it is a finite type two invariant. This theorem was proved in [19] and in [4]. Since the set up there was for closed knots, the diagrams were based on circles, not segments, and one hence had to include some factors in the above formula having to do with the automorphisms of those diagrams. We will encounter automorphism factors like these in Theorem 3.4.1. Recall the discussion of finite type invariants from Section 2.3. The invariant from Theorem 3.3.5 turns out to be the unique finite type two invariant which takes the value of zero on the unknot and one on the trefoil. It is also equal to the coeﬃcient of the quadratic term of the Conway polynomial, and it is equal to the Arf invariant when reduced mod 2. In addition, it appears in the surgery formula for the Casson invariant of homology spheres and is thus also known as the Casson knot invariant. A treatment of this invariant from the Casson point of view can be found in [42]. Proof of Theorem 3.3.5. In light of Propositions 3.3.1 and 3.3.4, the only thing to show is that the integrals along the principal faces match. This is clear essentially from the pictures (since the diagram pictures representing those collisions are the same), except that the collisions of K(xi ), i = 1, 2, 3, with x4 produce an extra map on each face (extension of Φi4 to that face) that is not present in the first integral. Since K(x4 ) can approach K(xi ) from any direction, the extension of Φi4 sweeps out a sphere, and this is independent of the other two maps. By Fubini’s Theorem, we then have for, say, the case i = 1, ! (Φ14 × Φ24 × Φ34 )∗ sym3S 2 ∂K(x1 )=x4 Conf[3,1;K,R3 ]

∗

(Φ14 ) symS 2 · !

(Φ24 × Φ34 )∗ sym2S 2

∂K(x1 )=x4 Conf[3,1;K,R3 ]

(Φ12 × Φ13 )∗ sym2S 2

Conf[3,R]

The last line is obtained by observing that the first integral in the previous line is 1 (since symS 2 is a unit volume form) and by rewriting the domain ∂K(x1 )=x4 Conf[3, 1; K, R3 ] as Conf[3, R] (and relabeling the

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points). The result is precisely the integral of one of the principal faces of . The remaining two principal faces can similarly be matched (IK3 ) up. Some care should be taken with signs, and we leave it to the reader to check those, keeping in mind that relabeling the vertices of a diagram may introduce a sign in the integral (this corresponds to permuting the copies of R and R3 and, since the dimensions of these spaces are odd, this in turn preserves or reverses the orientation of the fiber depending on the sign of the permutation). Changing orientations of chords of edges also might affect the sign (this corresponds to composing with the antipodal map to S 2 which changes the sign of the pullback form). To show that this is a finite type two invariant is not difficult. The key is that the resolutions of the three singularities can be chosen as small as desired. Then the integration domain can be broken up into subsets on which the difference of the integrals between the two resolutions is zero. Details can be found in [56, Section 5].

3.4

Finite type k knot invariants

Recall the space of trivalent diagrams from Definition 2.3.8. The two diagrams appearing in Theorem 3.3.5 are the two (up to decorations) elements of T D2 . However, the recipe for integration that these diagrams gave us in the previous section generalizes to any diagram. Namely, any diagram Γ ∈ T Dk with p segement vertices and q free vertices gives a prescription for constructing a pullback as in (11): (12)

Conf[p, q; K3 , R3 ]

! Conf[p + q, R3 ]

proj

Conf[p, R] × K3

eval

! Conf[p, R3 ]

There is again a bundle π : Conf[p, q; K3 , R3 ] −→ K3 whose fibers are manifolds with corners. We also have a map Φ : Conf[p, q; K3 , R3 ] −→ (S 2 )e where

Configuration space integrals and knots

• Φ is the product of the Gauss maps between pairs of configuration points corresponding to the edges of Γ, and • e is the number of chords and edges of Γ. Let α = Φ∗ (symeS 2 ). It is not hard to see that, because of the trivalence condition on T Dk , the dimension of the fiber of π is 2e, as is the dimension of α. Thus we get a 0-form π∗ α, or, in the notation of Section 3.3, a form (IK3 )Γ ∈ Ω0 (K3 ), whose value on K ∈ K3 is

(IK3 )Γ (K) =

α.

π −1 (K)=Conf[p,q;K,R3 ]

Now recall from discussion preceeding (7) that T W k is the space of weight systems of degree k, i.e. functionals on T Dk . Also recall the “self-linking” integral from equation (9). Finally let T B k be a basis of diagrams for T Dk (this is finite and canonical up to sign) and let | Aut(Γ)| be the number of automorphisms of Γ (these are automorphisms that fix the segment, regarded up to labels and edge orientations). Theorem 3.4.1 ([51]). For each W ∈ T W k , the map (13)

K3 −→ R $ " # W (Γ) (I 3 )Γ − µΓ A(K) , K $−→ | Aut(Γ)| K Γ∈T Bk

where µΓ is a real number that only depends on Γ, is a finite type k invariant. Furthermore, the assignment W $→ V ∈ Vk gives an isomorphism 0 IK 3 : T W k −→ Vk /Vk−1 (where the map (13) is followed by the quotient map Vk → Vk /Vk−1 ).

Proof. The argument here is essentially the same as in Theorem 3.3.5 but is complicated by the increased number of cases. Once again, to start, one should observe that the relations in Definition 2.3.8 are compatible with the sign changes that occur in the integral if copies of R or R3 in the bundle Conf[p, q; K3 , R3 ] are switched (the orientation of this space would potentially change and so would the sign of the integral) or if a Gauss map is replaced by its antipode.

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To prove that the integrals along hidden faces vanish, one considers various types of faces as in Propositions 3.3.1 and 3.3.4. If the points that are colliding form a “disconnected stratum” in the sense that the set of vertices and edges of the corresponding part of Γ forms two subsets such that no chord of edge connects a vertex of one subset to a vertex of the other, we revise the definition of Conf[p, q; K3 , R3 ] and turn this stratum into one of codimension greater than one. This takes care of most hidden faces [56, Section 4.2]. The remaining ones are disposed of by symmetry arguments due to Kontsevich [26] (see also [56, Section 4.3]) which show that some of the integrals are equal to their negatives and thus vanish. Another reference for the vanishing along hidden faces is [12, Theorem A.6] (the authors of that paper consider closed knots but this does not change the arguments). The vanishing of the integrals along faces at infinity goes exactly the same way as in Proposition 3.3.1; the map Conf[p, q; K3 , R3 ] → (S 2 )e always factors through a space of lower dimension. More details can be found in [56, Section 4.5]. Lastly, the vanishing of principal faces occurs due to the STU and IHX relations. The STU case is provided in Figure 13 (we have omitted the labels on diagrams and signs to simplify the picture). ! d W(

)(IK3 ) (K) ± W (

= W(

)d(IK3 ) (K) ± W (

= W(

)(IK3 ) (K) ± W (

! = W(

) ± W(

)(IK3 ) (K) ± W (

)(IK3 ) (K)

)d(IK3 ) (K) ± W (

)d(IK3 ) (K)

)(IK3 ) (K) ± W (

)(IK3 ) (K)

" ) (IK3 ) (K) = 0

Figure 13: Cancellation due to the STU relation Similar cancellation occurs with principal faces resulting from collision of free vertices, where one now uses the IHX relation. The contributions from all principal faces thus cancel. The one boundary integral that is not know to vanish (but is conjectured to; it is known that it does in a few low degree cases) is that of the anomalous face corresponding to all points colliding. It turns out that this integral is some multiple µΓ of the self-linking integral A(K),

Configuration space integrals and knots

and hence the term µΓ A(K) is subtracted so that we get an invariant. For further details, see [56, Section 4.6]. To show that this invariant is finite type k and that we get an isomorphism T W k → Vk /Vk−1 is tedious but straightforward. The point is that, as in the proof of Theorem 3.3.5, the resolutions of the k + 1 singularities can be chosen to diﬀer in arbitratily small neighborhoods and so the integrals from the sum of (13) cancel out. For this to happen, the domain of integration is subdivided and the integrals end up pairing oﬀ and canceling on appropriate neighborhoods. For details on how this leads to the conclusion that the invariant is finite type, see [56, Lemma 0 , when composed with 5.4]. Finally, it is easy to show that the map IK 3 the isomorphism (7), is an inverse to the map f from equation (6), and this gives the desired isomorphism [56, Theorem 5.3]. Remark 3.4.2. In combination with (7), Theorem 3.4.1 thus gives an alternative proof of Kontsevich’s Theorem 2.3.6. 0 Remark 3.4.3. The reason we put a superscript “0” on the map IK 3 is that this is really just the degree zero manifestation of a chain map described in Section 4.

Remark 3.4.4. In Theorem 3.3.5, there is one weight system for the degree two case and it takes one the values 1 and −1 for the two relevant diagrams. In addition, the anomalous correction in that case vanishes, so Theorem 3.4.1 is indeed a generalization of Theorem 3.3.5. Theorem 3.4.1 thus gives a way to construct all finite type invariants. In addition, configuration space integrals provide an important link between Chern-Simons Theory (where the first instances of such integrals occur), topology, and combinatorics. Unfortunately, computations with these integrals are diﬃcult and only a handful have been performed.

Generalization to Kn , n > 3

Just as there was no reason to stop at diagrams with four vertices, there is no reason to stop at trivalent diagrams. One might as well take diagrams that are at least trivalent, such as the one from Figure 14 (less than trivalent turns out not to give anything useful). To make this precise, we generalize Definition 2.3.7 as follows:

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Figure 14: A more general diagram (without decorations).

Definition 4.1. For n ≥ 3, define a diagram to be a connected graph consisting of an oriented line segment (considered up to orientationpreserving diffeomorphism) and some number of vertices of two types: segment vertices, lying on the segment, and free vertices, lying off the segment. The graph also contains some number of • chords connecting distinct segment vertices; • loops connecting segment vertices to themselves; and • edges connecting two free vertices or a free vertex and a segment vertex. Each vertex is at least trivalent, with the segment adding two to the count of the valence of a segment vertex. In addition, • if n is odd, all vertices are labeled, and edges, loops, and chords are oriented; • if n is even, external vertices, edges, loops, and chords are labeled. We also identify arcs, which are parts of the segment between successive segment vertices. Definition 4.2. For each n ≥ 3, let Dn be the real vector space generated by diagrams from Definition 4.1 modulo the relations 1. If Γ contains more than one edge connecting two vertices, then Γ = 0; 2. If n is odd and if a diagram Γ differs from Γ′ by a relabeling of vertices or orientations of loops, chord, and edges, then Γ − (−1)σ Γ′ = 0 where σ is the sum of the sign of the permutation of vertex labels and the number or loops, chords, and edges with different orientation.

Configuration space integrals and knots

3. If n is even and if a diagram Γ diﬀers from Γ′ by a relabeling of segment vertices or loops, chord, and edges, then Γ = (−1)σ Γ′ , where σ is the sum of the signs of these permutations. Note that T Dk is the quotient of the subspace of Dn generated by trivalent diagrams with 2k vertices. Definition 4.3. Define the degree of Γ ∈ Dn to be deg(Γ) = 2(# edges) − 3(# free vertices) − (# segment vertices). Thus if Γ is a trivalent diagram, deg(Γ) = 0. Coboundary δ is given on each diagram by contracting edges and segments (but not chords or loops since this does not represent a collision of points). Namely, let e be an edge or an arc of Γ and let Γ/e be Γ with e contracted. Then ! δ(Γ) = ϵ(Γ)Γ/e, edges and arcs e of Γ

where ϵ(Γ) is a sign determined by • if n is odd and e connects vertex i to vertex j, ϵ(Γ) = (−1)j if j > i and ϵ(Γ) = (−1)i+1 if j < i; • if n is even and e is an arc connecting vertex i to vertex j, then ϵ(Γ) is computed as above, and if e is an edge, then ϵ(Γ) = (−1)s , where s = (label of e) + (# segment vertices) + 1. On Dn , δ is the linear extension of this. An example (without decorations and hence modulo signs) is given in Figure 15. Theorem 4.4 ([12], Theorem 4.2). For n ≥ 3, (Dn , δ) is a cochain complex. Proof. It is a straightforward combinatorial exercise to see that δ raises degree by 1 and that δ 2 = 0. Returning to configuration space integrals, for each Γ ∈ Dn and K ∈ Kn , we can still define an integral as before. Vertices of Γ tell

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(last two are zero)

Figure 15: An example of a coboundary.

us what pullback bundle Conf[p, q; Kn , Rn ] to construct, i.e. how many points to have on and oﬀ the knot. The only diﬀerence is that the map Φ : Conf[p, q; Kn , Rn ] −→ (S n−1 )(#

loops, chords, and edges of Γ)

is now a product of Gauss maps for each edge and chord of Γ as well as the derivative map for each loop of Γ. Further, none of what was done in Section 3.4 requires n = 3. More precisely, we still get a form, for n ≥ 3, (IKn )Γ ∈ Ωd (Kn ), given by integration along the fiber of the bundle π : Conf[p, q; Kn , Rn ] −→ Kn . The degree of the form is no longer necessarily zero but of degree equal to (n − 1)(# loops, chords, and edges of Γ)

−n(# free vertices of Γ)

−(# segment vertices of Γ) and its value on K ∈ Kn is as before (IKn )Γ (K) =

α.

π −1 (K)=Conf[p,q;K,Rn ]

Theorem 4.5 ([12], Theorem 4.4). For n > 3, configuration space integrals give a cochain map IKn : (Dn , δ) −→ (Ω∗ (Kn ), d).

Configuration space integrals and knots

(Here d is the ordinary deRham differential.) Proof. To prove this, one first observes that, just as in Theorem 3.4.1, the relations in Dn correspond precisely to what happens on the integration side if points or maps are permuted or if Gauss maps are composed with the antipodal map. In fact, those relations in Dn are defined precisely because of what happens on the integration side. Once this is established, it is necessary to show that, for each Γ, the integrals along the hidden faces and faces at infinity vanish, and this goes exactly the same way as in Theorem 3.4.1. The integrals along principal faces correspond precisely to contractions of edges and arcs, so that the map commutes with the differential. Remark 4.6. There is an algebra structure on Dn given by the shuffle product that is compatible with the wedge product of forms [13]. This makes IKn a map of algebras as well. By evaluating IKn on certain diagrams, Cattaneo, Cotta-Ramusino, and Longoni [12] also prove Corollary 4.7. Given any i > 0, the knot space Kn , n > 3, has nontrivial cohomology in dimension greater than i. Complex Dn is known to have the same homology as the Kn , so it contains a lot of information about the topology of long knots. However, we do not know that this map induces an isomorphism. More precisely, we have Conjecture 4.8. The map IKn is a quasi-isomorphism. Even though we do not have Theorem 4.5 for n = 3, the construction is compatible with what we already did in the case of classical knots K3 . Namely, for n = 3, one does not get a cochain map in all degrees because of the anomalous face. But in degree zero, it turns out that H0 (D3 ) = T D (up to certain diagram automorphism factors; see [30, Section 3.4]). In other words, the kernel of δ in degree zero is defined by imposing the 1T, STU, and IHX relations. Thus, after correcting by the anomalous correction and after identifying T D with its dual, the space of weight systems T W (the dualization gymnastics is described in [30, Section 3.4]), we get a map (H0 (D3 ))∗ = T W −→ H0 (K).

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0 from Theorem 3.4.1 and we already But this is precisely the map IK 3 know that the image of this map is the finite type knot invariants.

Further generalizations and applications

In this section we give brief overviews of other contexts in which configuration space integrals have appeared in recent years.

5.1

Spaces of links

Configuration space integrals can also be defined for spaces of long links, homotopy links, and braids. (The reader should keep in mind that it is actually quite surprising that they can be defined for homotopy links.) The results stated here encompass those for knots (by setting m = 1 in Lnm ). For m ≥ 1, n ≥ 2, let Mapc ("m R, Rn ) be the space of smooth maps of "m R to Rn which, outside of "m I agree with the map "m R → Rn , which is on the ith copy of R given as t $−→ (t, i, 0, 0, ..., 0). As in the case of knots (Definition 2.2.1), we can define the spaces of links as subspaces of Mapc ("m R, Rn ) with the induced topology as follows. Definition 5.1.1. Define the space of • long (or string) links with m strands Lnm ⊂ Mapc ("m R, Rn ) as the space of embeddings L : "m R → Rn . n ⊂ Map (" R, Rn ) as the space of • pure braids on m strands Bm m c n embeddings B : "m R → R whose derivative in the direction of the first coordinate is positive. n ⊂ Map (" R, • long (or string) homotopy links with m strands Hm m c n n R ) as the space of link maps H : "m R → R (smooth maps of "m R in Rn with the images of the copies of R disjoint).

Remark 5.1.2. Another (and in fact, more standard) way to think n is as the loop space Ω Conf(m, Rn−1 ). about Bm

Configuration space integrals and knots

Remark 5.1.3. For technical reasons, it is sometimes better to take strands that are not parallel outside of I n , but this does not change anything about the theorems described here. For details, see [30, Definition 2.1]. Some observations about these spaces are: n ⊂ Ln ⊂ H n ; • Bm m m n ), we can pass a strand through itself so this can be • In π0 (Hm thought of as space of “links without knotting”;

Example of a homotopy link and a braid is given in Figure 16. Note that, as usual, we have confused the maps H and L with their images in Rn .

H ∈ H3n self-intersection

n L ∈ B3n ⊂ Ln 3 ⊂ H3

Figure 16: Examples of links. The top picture is a homotopy link, but not a link (and hence not a braid) because of the self-intersection in the bottom strand. When we say “link”, we will mean an embedded link. Otherwise we will say “homotopy link” or “braid”. As with knots, the adjective “long” will be dropped. We will denote components (i.e. strands) of an embedded link by L = (K1 , K2 , ..., Km ). As before, an isotopy is a homotopy in the space of links or braids, and link homotopy is a path in the space of homotopy links. As in the case of the space of knots, all of these link spaces are smooth infinite-dimensional paracompact manifolds so we can consider their deRham cohomology.

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Finite type invariants of these link spaces can be defined the same way as for knots (see Section 2.3). Namely, we consider self-intersections which, in the case of links, come from a single strand or two diﬀerent strands (i.e. in the left picture of Figure 2, there are no conditions on the two strands making up the singularity). For homotopy links, we only take intersections that come from diﬀerent strands. For braids, this condition is automatic since a braid cannot “turn back” to intersect itself. Then a finite type k invariant is defined as an invariant that vanishes on (k + 1) self-intersections. We will denote finite type k invariants of links, homotopy links, and braids by LV k , HV k , and BV k , respectively. As for knots, the question of separation of links by finite type invariants is still open, but it is known that these invariants separate homotopy links [20] and braids [6, 24]. We now revisit Section 4 and show how Theorem 4.5 generalizes to links. Namely, recall the cochain map IKn : Dn −→ Ω∗ (Kn ). The first order of business is to generalize the diagram complex Dn to a complex LDnm (which we will use for both links and braids) and a subcomplex HDnm (which we will use for homotopy links). This generalization is simple: LDnm is defined the same way as Dn except there are now m segments, as for example in Figure 17.

∈ LDn4 Figure 17: An example of a diagram for links (without decorations). All the definitions from Section 4 carry over in exactly the same way and we will not reproduce the details here, especially since they are spelled out in [30, Section 3]. In particular, depending on the parity of n, the diagrams have to be appropriately decorated. The diﬀerential is again given by contracting arcs and edges. HDnm is defined by taking diagrams

Configuration space integrals and knots

• with no loops, and • requiring that, if there exists a path between distinct vertices on a given segment, then it must pass through a vertex on another segment. It is a simple combinatorial exercise to show that HDnm is a subcomplex of LDnm [30, Proposition 3.24]. n are still defined As expected, in degree zero, complexes Lnm and Hm by imposing the STU and IHX relations, and an extra relation in the n that diagrams cannot contain closed paths of edges. In case of Hm particular, the spaces of weight systems of degree k for these link spaces, which we will denote by LW k and HW k , consist of functionals vanishing on these relations (with automorphism factors); see [30, Definition 3.35] for details.

As it turns out, the integration is not as easily generalized. The n , then problem is that, if we want to produce cohomology classes on Hm n the evaluation map from (12) will need to take values in Hm , but points in this space are not even immersions, let alone embeddings. Hence the target of the evaluation map would not be a configuration space but rather some kind of a “partial configuration space” where some points are allowed to pass through each other (this is actually a complement of a subspace arrangement, a familiar object from algebraic geometry). But then the projection map would be a map of partial configuration spaces which is far from being a fibration (see [30, Example 4.7]). This n. makes it unlikely that the pullback is a bundle over Hm A way around this is to patch the integral together from pieces for which this diﬃculty does not occur. This is achieved by breaking up a diagram Γ ∈ LDnm into its graft components which are essentially the components one would see after the segments and segment vertices are removed. The second condition defining the subcomplex HDnm guarantees that there will be no more than one segment vertex on each segment of each graft, and this turns out to remove the issue of the projection not being a fibration. Since it would take us too far afield to define the graft-based pullback bundle precisely, we will refer the reader to [30, Definition 4.16] for details. Suﬃce it to say here that the construction essentially takes into account both the vertices and edges of Γ rather than just vertices when constructing the pullback bundle (see Remark 3.3.2). This procedure is indeed a refinement of the original definition

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of configuration space integrals since it produces the same form on Lnm as the original definition [30, Proposition 4.24]. The only diﬀerence, therefore, is that the graft definition makes it possible to restrict the integration from the complex LDnm to the subcomplex HDnm and produce n. forms on Hm We then have a generalization of Theorem 4.5. Theorem 5.1.4 ([30], Section 4.5). There are integration maps IL and IH given by configuration space interals and a commutative diagram HD! nm

! Ω∗ (Hn ) m

" ! Ω∗ (Ln ) m

LDnm

Here IL is a cochain map for n > 3 and IH is a cochain map for n ≥ 3. Proof. The proof goes exactly the same way as in the case of knots. The only difference is that IH is also a cochain map for n = 3. The reason for this is that the anomalous face is not an issue for homotopy links. Namely, the anomaly can only arise when all points on and off the link collide. But since strands of the link are always disjoint, this is only possible when all the configuration points on the link are in fact on a single strand. In other words, the corresponding diagram Γ is concentrated on a single strand. Such a diagram does not occur in HDnm . (The integral in this case in effect produces a form on the space of knots, so that the anomaly can be thought of as a purely knotting, rather than linking, phenomenon.) Remark 5.1.5. As in the case of knots, there is an algebra structure on LDnm given by the shuffle product that is compatible with the wedge product of forms [30, Section 3.3.2]. It thus turns out that the maps IL and IH are maps of algebras [30, Proposition 4.29]. For n = 3, we also have a generalization of Theorem 3.4.1. Theorem 5.1.6 ([30], Theorems 5.6 and 5.8). Configuration space integral maps IL and IH induce isomorphisms ∼ =

IL0 : LW k −→ LV k /LV k−1 ⊂ H0 (L3m ) ∼ =

0 3 : HW k −→ LV k /LV k−1 ⊂ H0 (Hm ) IH

Configuration space integrals and knots

The isomorphisms are given exactly as in Theorem 3.4.1. In particular, the anomalous correction has to be introduced for the case of links. Lastly, we mention an interesting connection to a class of classical homotopy link invariants called Milnor invariants [36]. In brief, these invariants live in the lower central series of the link group and essentially measure how far a “longitude” of the link survives in the lower central series. It is known that Milnor invariants of long homotopy links are finite type invariants (and it is important that these are long, rather than closed homotopy links) [6, 32]. Theorem 5.1.6 thus immediately gives us Corollary 5.1.7. The map IH provides configuration space integral ex3 . pressions for Milnor invariants of Hm For more details about this corollary, see [30, Section 5.4]. Even though we made no explicit mention of braids in the above theorems, everything goes through the same way for this space as well. The complex is still LDnm but the evaluation now take place on braids, n . The integration I would thus produce forms on B n i.e. elements of Bm B m 3 . However, this is not very satisfying and all finite type invariants of Bm since we do not yet have a good subcomplex of LDnm or a modification in the integration procedure that would take into account the definition n ≃ Ω Conf(m, Rn−1 ), braids can be of braids. For example, since Bm thought of as “flowing” at the same rate, and the integration hence might be defined so that it only takes place in “vertical slices” of the braid. In particular, one should be able to connect configuration space integrals for braids to Kohno’s work on braids and Chen integrals [24].

5.2

Manifold calculus of functors and finite type invariants

Configuration space integrals connect in unexpected ways to the theory of manifold calculus of functors [59, 18]. Before we can state the results, we provide some basic background, but we will assume the reader is familiar with the language of categories and functors. For further details on manifold calculus of functors, the reader might find [40] useful. For M a smooth manifold, Let Top be the category of topological spaces and let O(M ) = category of open subsets of M with inclusions as morphisms.

Ismar Voli´c

Manifold calculus studies functors F : O(M )op −→ Top satisfying the conditions: 1. F takes isotopy equivalences to homotopy equivalences, and; 2. For any sequence of open sets U0 ⊂ U1 ⊂ · · · , the canonical map F (∪i Ui ) → holimi F (Ui ) is a homotopy equivalence (here “holim” stands for the homotopy limit). The target category is not limited to topological spaces, but for concreteness and for our purposes we will stick to that case. One such functor is the space of embeddings Emb(−, N ), where N is a smooth manifold, since, given an inclusion O1 !→ O2 of open subsets of M , there is a restriction Emb(O2 , N ) −→ Emb(O1 , N ). In particular, we can specialize to the space of knots Kn , n ≥ 3, and see what manifold calculus has to say about it. For any functor F : O(M )op → Top, the theory produces a “Taylor tower” of approximating functors/fibrations ! " F (−) −→ T0 F (−) ← · · · ← Tk F (−) ← · · · ← T∞ F (−) where T∞ F (−) is the inverse limit of the tower.

Theorem 5.2.1 ([60]). For F = Emb(−, N ) and for 2 dim(M ) + 2 ≤ dim(N ), the Taylor tower converges on (co)homology (for any coeﬃcients), i.e. H∗ (Emb(−, N )) ∼ = H∗ (T∞ Emb(−, N )). In particular, evaluating at M gives H∗ (Emb(M, N )) ∼ = H∗ (T∞ Emb(M, N )). Remark 5.2.2. For dim(M ) + 3 ≤ dim(N ), the same convergence result is true on homotopy groups [17].

Configuration space integrals and knots

Note that when M is 1-dimensional, N has to be at least 4-dimensional to guarantee convergence. Hence this says nothing about K3 . Nevertheless, the tower can still be constructed in this case and it turns out to contain a lot of information. To construct Tk Kn , n ≥ 3, let I1 , ..., Ik+1 be disjoint closed subintervals of R and let ∅ ̸= S ⊆ {1, ..., k + 1}. Then let KSn = Embc (R \

Ii , Rn ),

i∈S

where Embc as usual stands for the space of “compactly supported” embeddings, namely those that are fixed outside some compact set such as I. Thus KSn is a space of “punctured knots”; an example is given in Figure 18.

n Figure 18: An element of K{1,2,3,4} .

These spaces are not very interesting on their own, and are connected n even for n = 3. However, we have restriction maps KSn → KS∪{i} given by “punching another hole”, namely restricting an embedding of R with some intervals taken out to an embedding of R with one more interval taken out. These spaces and maps then form a diagram of knots with holes (such a diagram is sometimes called a punctured cube). Example 5.2.3. When k = 2, we get n K{1}

" # Kn

n K{2} n K{3}

{1,2}

# Kn

{1,3}

n K{2,3}

" ! # Kn {1,2,3}

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Definition 5.2.4. The kth stage of the Taylor tower for Kn , n ≥ 3, is the homotopy limit of this punctured cube, i.e. Tk K n =

holim

∅̸=S⊆{1,..,k+1}

KSn .

For the reader not familiar with homotopy limits, it is actually not hard to see what this homotopy limit is: For example, the punctured cubical diagram from Example 5.2.3 can be redrawn as n K{1}

n K{1,3}

n K{3}

# Kn $ {1,2,3} &

n K{1,2}

$ # Kn {2,3}

n K{2}

Then a point in T2 Kn is n (once-punctured knot); • A point in each K{i} n • A path in each K{i,j} (isotopy of a twice-punctured knot) ; n • A two-parameter path in K{1,2,3} (two-parameter isotopy of a thrice-punctured knot); and

• Everything is compatible with the restriction maps. Namely, a n n path in each K{i,j} joins the restrictions of the elements of K{i} n n n and K{j} to K{i,j} , and the two-parameter path in K{1,2,3} is ren which, on its edges, is the ally a map of a 2-simplex into K{1,2,3} n n . restriction of the paths in K{i,j} to K{1,2,3} The pattern for Tk Kn , k ̸= 2, should be clear. There is a map Kn −→ Tk Kn given by punching holes in the knot (the isotopies in the homotopy limit are thus constant). Remark 5.2.5. It is not hard to see that, for k ≥ 3, Kn is the actual pullback (limit) of the subcubical diagram. So the strategy here is to replace the limit, which is what we really care about, by the homotopy limit, which is hopefully easier to understand.

Configuration space integrals and knots

There is also a map, for all k ≥ 1, Tk Kn −→ Tk−1 Kn , since the diagram defining Tk−1 Kn is a subdiagram of the one defining Tk Kn (this map is a fibration; this is a general fact about homotopy limits). Putting these maps and spaces together, we get the Taylor tower for n ≥ 3: " ! Kn −→ T0 Kn ← · · · ← Tk Kn ← · · · ← T∞ Kn .

Kn ,

By Theorem 5.2.1, this tower converges on (co)homology for n ≥ 4.

There is a variant of this Taylor tower, the so-called “algebraic Taylor tower”, which is a tower of cochain complexs obtained by applying cochains to each space of punctured knots and then letting Tk∗ (K3 ) be the homotopy colimit of the resulting diagram of cochain complexes. 0 from Theorem 3.4.1. We then have the following Recall the map IK 3 theorem, which essentially says that the algebraic Taylor tower classifies finite type invariants. 0 factors through the Theorem 5.2.6 ([55], Theorem 1.2). The map IK 3 3 algebraic Taylor tower for K . Furthermore, we have isomorphisms

T Wk

I0 3 K

∼ =

! Vk /Vk−1 #

∗ K3 ) H0 (T2k ∗ K3 ) ∼ H0 (T ∗ 3 (and H0 (T2k = 2k+1 K ) so all stages are accounted for).

The main ingredient in this proof is the extension of configuration space integrals to the stages T2k K3 of the space of long knots [54, Theorem 4.5]. The idea of this extension is this: As a configuration point moves around a punctured knot (this corresponds to a point moving on the knot in the usual construction) and approaches a hole, it is made to “jump”, via a path in the homotopy limit (this is achieved by an appropriate partition of unity), to another punctured knot which has that hole filled in, thus preventing the evaluation map from being undefined.

Ismar Voli´c

Theorem 5.2.6 places finite type theory into a more homotopy-theoretic setting and the hope is that this might give a new strategy for proving the separation conjecture. Several generalizations of Theorem 5.2.6 should be possible. For example, it should also be possible to extend the entire chain map from Theorem 4.5 to the Taylor tower. It should also be possible to show that n , and B n supplied by the multivarithe Taylor multitowers for Lnm , Hm m able manifold calculus of functors [41] also admit factorizations of the integration maps IL , IH , and IB , as well as classify finite type invariants of these spaces. Lastly, it seems likely that finite type invariants are all one finds in the ordinary Taylor tower (and not just its algebraic version). Some progress toward this goal can be found in [11].

5.3

Formality of the little balls operad

There is another striking connection between the Taylor tower for knots and configuration space integrals of a slightly different flavor. To explain, we will first modify the space of knots slightly and instead of Kn , n > 3, use the space Kn = hofiber(Embc (R, Rn ) !→ Immc (R, Rn )), where hofiber stands for the homotopy fiber. This is the space of “embeddings modulo immersions” and a point in it is a long knot along with a path, i.e. a regular homotopy, to the long unknot (since this is a natural basepoint in the space of immersions) through compactly supported immersions. This space is easier to work with, but it is not very different from Kn : the above inclusion is nullhomotopic [46, Proposition 5.17] so that we have a homotopy equivalence Kn ≃ Kn × Ω2 S n−1 . The difference between long knots and its version modulo immersions is thus well-undersrtood, especially rationally. Now let Bn = {Bn (p)}p≥0 be the little n-discs operad (i.e. little balls in Rn ), where Bn (p) is the space of configurations of p closed n-discs with disjoint interiors contained in the unit disk of Rn . The little ndiscs operad is an important object in homotopy theory, and a good introduction for the reader who is not familiar with it, or with operads

Configuration space integrals and knots

in general, is [39]. Taking the chains and the homology of Bn gives two operads of chain complexes, C∗ (Bn ; R) and H∗ (Bn ; R) (where the latter is a chain complex with zero diﬀerential). Then we have the following formality theorem. Theorem 5.3.1 (Kontsevich [28]; Tamarkin for n = 2 [50]). For n ≥ 2, there exists a chain of weak equivalences of operads of chain complexes ≃

≃

!n −→ H∗ (Bn ; R), C∗ (Bn ; R) ←− D

!n is a certain diagram complex. In other words, Bn is (stably) where D formal over R.

For details about the proof of Theorem 5.3.1, the reader should con!n is a diagram sult [35]. The reason this theorem is relevant here is that D complex (it is in fact a commutative diﬀerential graded algebra cooperad) that is essentially the complex Dn we encountered in Section 4. !n . In addition, The main diﬀerence is that loops are not allowed in D the map !n −→ C∗ (Bn ; R) D

is given by configuration space integrals and is the same as the map IKn we saw before, with one important diﬀerence that the bundle we integrate over is diﬀerent. To explain briefly, Bn can be regarded as a !n collection of configuration spaces Conf[p, Rn ]. For a diagram Γ ∈ D with p segment vertices and q free vertices, consider the projection π : Conf[p + q, Rn ] −→ Conf[p, Rn ].

This is a bundle in a suitable sense; see [35, Section 5.9]. The edges of Γ again determine some Gauss maps Conf[p + q, Rn ] → S n−1 , so that the product of volume forms can be pulled back to Conf[p + q, Rn ] and then pushed forward to Conf[p, Rn ]. This part is completely analogous to what we have seen in Section 4. The bulk of the proof of Theorem 5.3.1 !n → Ω∗ (Conf[p, Rn ]) is an equivthen consist of showing that the map D !n and its dual, as alence (we are liberally passing between cooperad D well as chains and cochains on configuration spaces). This again comes down to vanishing arguments, which are the same as in Theorem 4.5. The map

!n −→ H∗ (Bn ; R) D

is easy, with some obvious diagrams sent to the generators of the (co)homology of configuration spaces and the rest to zero.

Ismar Voli´c

Theorem 5.3.1 has been used in a variety of situations, such as McClure-Smith’s proof of the Deligne Conjecture and Tamarkin’s proof of Kontsevich’s deformation quantization theorem. For us, the importance is in that it gives information about the rational homology of Kn , n > 3. To explain, first observe that the construction of the Taylor tower for Kn from Section 5.2 can be carried out in exactly the same way for Kn . Then, by retracting the arcs of a punctured knot, we get KSn ≃ Conf(|S| − 1, Rn ). If we had used Kn , we would also have copies of spheres keeping track of tangential data. In the Kn version, they are not present since the space of immersions, which carries this data, has been removed. The restriction maps “add a point”, as in Figure 19. x1

x2 x3

↓ x′2 x3

Figure 19: Restriction of punctured knots. This structure yields a homology spectral sequence that can be associated to the Taylor tower for Kn , n ≥ 3. It starts with 1 E−p,q = Hq (Conf(p, Rn )),

and, for n ≥ 4, converges to H∗ (T∞ Kn ). By Theorem 5.2.1, this spectral sequence hence converges to H∗ (Kn ). Remark 5.3.2. This is the Bousfield-Kan spectral sequence that can be associated to any cosimplicial space, and in particular to the cosimplicial space defined by Sinha [47] that models the Taylor tower for Kn . The spaces in this cosimplicial model are slight modifications of the FultonMacPherson compactification of Conf(p, Rn ) and the maps “double” and “forget” points. In particular, the doubling maps are motivated by

Configuration space integrals and knots

the situation from Figure 19. In addition, this turns out to be the same spectral sequence (up to regrading) as the one defined by Vassiliev [53] which motivated the original definition of finite type knot invariants. Theorem 5.3.3. The homology spectral sequence described above collapses rationally at the E 2 page for n ≥ 3. This theorem was proved for n ≥ 4 in [34], for n = 3 on the diagonal in [27], and for n = 3 everywhere in [37] and [48]. Theorem 5.2.6 can also be interpreted as the collapse of this spectral sequence on the diagonal for n = 3. Idea of proof. The vertical differential in the spectral sequence is the ordinary internal differential on the cochain complexes of configuration spaces (the vertical one has to do with doubling configuration points, and this has to do with Figure 19). By Theorem 5.3.1, this differential can be replaced by the zero differential, and hence the spectral sequence collapses. Some more sophisticated model category theory techniques are required for the case n = 3. Remark 5.3.4. Collapse is also true for the homotopy spectral sequence for n ≥ 4 [1]. So for n ≥ 4, the homology of the E 2 page is the homology of Kn . A more precise way to say this is Theorem 5.3.5. For n ≥ 4, H∗ (Kn ; Q) is the Hochschild homology of the Poisson operad of degree n−1, which is the operad obtained by taking the homology of the little n-cubes operad. For more details on the Poisson operad of degree n − 1, see [52, Section 1]. Briefly, this is the operad encoding Poisson algebras, i.e. graded commutative algebras with Lie bracket of degree n − 1 which is a graded derivation with respect to the multiplication. One can define a differential (the Gerstenhaber bracket), and the homology of the resulting complex is the Hochschild homology of the Poisson operad. In summary, H∗ (Kn ; Q) is built out of H∗ (Conf(p, Rn ); Q), p ≥ 0, which is understood. In fact, this homology can be represented combinatorially with graph complexes of chord diagrams. This therefore gives a nice combinatorial description of H∗ (Kn ; Q), n ≥ 4. The case n = 3 is not yet well understood, and the implications of the collapse are yet to be studied. The main impediment is that, even though the spectral

Ismar Voli´c

sequence collapses, it is not clear what the spectral sequence converges to. It would be nice to rework the results described in this section for spaces of links. For ordinary embedded links and braids, things should work the same, but homotopy links are more challenging since we do not yet have any sort of a convergence result for the Taylor (multi)tower for this space. Acknowledgements The author would like to thank Sadok Kallel for the invitation to write this article and the referee for a careful reading. Ismar Voli´c Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481, ivolic@wellesley.edu

References [1] Arone G.; Lambrechts P.; Turchin V.; Voli´c I., Coformality and rational homotopy groups of spaces of long knots, Math. Res. Lett. 15:1 (2008), 1–14. [2] Arone G.; Turchin V., On the rational homology of high dimensional analogues of spaces of long knots, arXiv:1105.1576 [math.AT] (26 Mar 2012). [3] Axelrod S.; Singer I. M., Chern-Simons perturbation theory, II, J. Diﬀerential Geom. 39:1 (1994), 173–213. [4] Bar-Natan D., Perturbative aspects of the Chern-Simons topological quantum field theory, Thesis Princeton Univ., 1991. [5] Bar-Natan D., On the Vassiliev knot invariants, Topology 34:2 (1995), 423–472. [6] Bar-Natan D., Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4:1 (1995), 13–32. [7] Birman J. S.; Lin X. -S., Knot polynomials and Vassiliev’s invariants, Invent. Math. 111:2 (1993), 225–270.

Configuration space integrals and knots

[8] Bott R.; Tu L. W., Diﬀerential Forms in Algebraic Topology, Graduate Text in Mathematics 82, Springer, New York, 1992. [9] Bott R.; Taubes C., On the self-linking of knots, J. Math. Phys. 35:10 (1994), 5247–5287. [10] Budney R., Little cubes and long knots, Topology 46:1 (2007), 1–27. [11] Budney R.; Conant J.; Scannell K. P.; Sinha D., New perspectives on self-linking, Adv. Math. 191:1 (2005), 78–113. [12] Cattaneo A. S.; Cotta-Ramusino P.; Longoni R., Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002), 949–1000. [13] Cattaneo A. S.; Cotta-Ramusino P.; Longoni R., Algebraic structures on graph cohomology, J. Knot Theory Ramifications 14:5 (2005), 627–640. [14] Cattaneo A. S.; Rossi C. A., Wilson surfaces and higher dimensional knot invariants, Comm. Math. Phys. 256:3 (2005), 513– 537. [15] Chmutov S.; Duzhin S.; Mostovoy J., Introduction to Vassiliev Knot Invariants, Cambridge University Press, Cambridge, 2012. [16] Fulton W., MacPherson R., A compactification of configuration spaces, Ann. of Math. II 139:1 (1994), 183–225. [17] Goodwillie T. G.; Klein J. R., Multiple disjunction for spaces of smooth embeddings, In preparation. [18] Goodwillie T. G.; Weiss M., Embeddings from the point of view of immersion theory II, Geom. Topol. 3 (1999), 103–118. [19] Guadagnini E.; Martellini M.; Mintchev M., Chern-Simons field theory and link invariants, In: Knots, Topology and Quantum Field Theories, Proceedings of the Johns Hopkins Workshop on Current Problems in Particle Theory, Florence (1989), World Scientific 13, 95–145. [20] Habegger N.; Lin X. -Song, The classification of links up to linkhomotopy, J. Amer. Math. Soc. 3:2 (1990), 389–419.

Ismar Voli´c

[21] Halperin S.; Greub W. H., Fibre integration and some of its applications, Collect. Math. 23 (1972), 195–212. [22] Hirsch M. W., Differential Topology, Graduate Texts in Mathematics, Springer-Verlag, New York, 1994. [23] Joyce D., On manifolds with corners, arXiv:0910.3518 [math.DG] (13 Oct 2010). [24] Kohno T., Loop spaces of configuration spaces and finite type invariants, In: Invariants of knots and 3-manifolds, Kyoto (2001), Geometry & Topology Monographs, 4 2002, 143–160. [25] Komendarczyk, R.; and Volić, I., On volume-preserving vector fields and finite type invariants of knots, In preparation. [26] Kontsevich M., Vassiliev’s knot invariants, In: I. M. Gelfand Seminar, Adv. Soviet Math., 16 1993, 137–150. [27] Kontsevich M., Feynman diagrams and low-dimensional topology, In: First European Congress of Mathematics, Paris (1992), Progr. Math., 120 1994, 97–121. [28] Kontsevich M., Operads and motives in deformation quantization, Lett. Math. Phys. 48:1 (1999), 35–72. [29] Koytcheff R., A homotopy-theoretic view of Bott-Taubes integrals and knot spaces, Algebr. Geom. Topol. 9:3 (2009), 1467–1501. [30] Koytcheff R.; Munson B. A.; Volić I., Configuration space integrals and the cohomology of the space of homotopy string links, J. Knot Theory Ramifications, to appear. [31] Lee J. M., Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer, New York, 2003. [32] Lin X. -S., Power series expansions and invariants of links, Geometric Topology, Athens GA (1993), AMS/IP Stud. Adv. Math. 2 1997, 184–202. [33] Longoni R., Nontrivial classes in H ∗ (Imb(S 1 , Rn )) from nontrivalent graph cocycles, Int. J. Geom. Methods Mod. Phys. 1:5 (2004), 639–650.

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[34] Lambrechts P.; Turchin V.; Volić I., The rational homology of spaces of long knots in codimension > 2, Geom. Topol. 14 (2010), 2151–2187. [35] Lambrechts P.; Volić I., Formality of the little N -discs operad, Memoirs of the AMS, to appear. [36] Milnor J., Link groups, Ann. of Math. II 59 (1954), 177-195. [37] Moriya S., Sinha’s spectral sequence and homotopical algebra of operads, arXiv:1210.0996 [math.AT] (3 Oct 2012). [38] Moskovich D., Framing and the self-linking integral, Far East J. Math. Sci. 14:2 (2004), 165–183. [39] Markl M.; Shnider S.; Stasheff J, Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs 96, American Mathematical Society, 2002. [40] Munson B. A., Introduction to the manifold calculus of GoodwillieWeiss, Morfismos 14:1 (2010), 1–50. [41] Munson B. A.; Volić I., Multivariable manifold calculus of functors, Forum Math. 24:5 (2012), 1023–1066. [42] Polyak M.; Viro O., On the Casson knot invariant, J. Knot Theory Ramifications 10:5 (2001), 711–738. [43] Sakai K., Nontrivalent graph cocycle and cohomology of the long knot space, Algebr. Geom. Topol. 8:3 (2008), 1499–1522. [44] Sakai K., Configuration space integrals for embedding spaces and the Haefliger invariant, J. Knot Theory Ramifications 19:12 (2010), 1597–1644. [45] Sinha D. P., Manifold-theoretic compactifications of configuration spaces, Selecta Math. 10:3 (2004), 391–428. [46] Sinha D. P., Operads and knot spaces, J. Amer. Math. Soc. 19:2 (2006), 461–486. [47] Sinha D. P., The topology of spaces of knots: cosimplicial models, Amer. J. Math. 131:4 (2009), 945–980.

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[48] Songhafouo-Tsopméné P. A., Formality of Sinha’s cosimplicial model for long knots spaces, arXiv:1210.2561 [math.AT] (9 Oct 2012). [49] Sakai K.; Watanabe T., 1-loop graphs and configuration space integral for embedding spaces, Math. Proc. Cambridge Philos. Soc. 152:3 (2012), 497–533. [50] Tamarkin D. E., Formality of chain operad of little discs, Lett. Math. Phys. 66:1–2 (2003), 65–72. [51] Thurston D., Integral expressions for the Vassiliev knot invariants, arXiv:math/9901110 [math.QA] (25 Jan 1999). [52] Tourtchine V., On the homology of the spaces of long knots, In: Advances in topological quantum field theory, NATO Sci. Ser., 179 (2004), 23–52. [53] Vassiliev V. A., Cohomology of knot spaces, In: Theory of singularities and its applications, Adv. Soviet Math., 1 (1990), 23–69. [54] Volić I., Configuration space integrals and Taylor towers for spaces of knots, Topology Appl. 153:15 (2006), 2893–2904. [55] Volić I., Finite type knot invariants and the calculus of functors, Compos. Math. 142:1 (2006), 222–250. [56] Volić I., A survey of Bott-Taubes integration, J. Knot Theory Ramifications 16:1 (2007), 1–42. [57] Volić I., On the cohomology of spaces of links and braids via configuration space integrals, Sarajevo J. Math. 6:19,2 (2010), 241–263. [58] Watanabe T., Configuration space integral for long n-knots and the Alexander polynomial, Algebr. Geom. Topol. 7 (2007), 47–92. [59] Weiss M., Embeddings from the point of view of immersion theory I, Geom. Topol. 3 (1999), 67–101. [60] Weiss M., Homology of spaces of smooth embeddings, Q. J. Math. 55:4 (2004), 499–504.

Morfismos, Vol. 17, No. 2, 2013, pp. 57–69 Morfismos, Vol. 17, No. 2, 2013, pp. 57–69

Topological chiral homology and configuration spaces of spheres Oscar Randal-Williams

Abstract We compute the rational homology of all spaces of finite configurations on spheres. Our tool is a bar spectral sequence which can be viewed as coming from the notion of “topological chiral homology”, though we give a self-contained construction of the spectral sequence.

2010 Mathematics Subject Classification: 55R80, 55T05, 57T30. Keywords and phrases: configuration space, spectral sequence.

Result

For a topological manifold M , let Cn (M ) denote the space of n unordered distinct points in M . The purpose of this short note is to give a proof of the following. Theorem 1.1. Suppose d ≥ 2 is even. Then ⎧ ⎪ ⎨Q in degree 2d − 1 d ! ∗ (Cn (S ); Q) = Q in degree d H ⎪ ⎩ 0 Suppose d ≥ 3 is odd. Then

! ∗ (Cn (S d ); Q) = H

Q in degree d 0

n≥3 n=1 n = 0, 2.

n≥1 n = 0.

Supported by the Herchel Smith Fund, ERC Advanced Grant No. 228082, and the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation.

O. Randal-Williams

This theorem is not new (for example, it may be quickly deduced from [7, Theorem 18]), but the method of proof we oﬀer is diﬀerent and, we feel, somewhat interesting.

A “multiplicative” decomposition of configuration spaces

Let Cn (M, X) denote the space of n unordered points in a topological manifold M , labeled by the space X, and let ! Cn (M, X). C(M, X) := n≥0

If the manifold M has dimension d, fix an embedding e : Rd !→ M and let S(t) ⊂ M denote the image under e of the sphere of radius t centred at the origin of Rd . Define Bn to be # " (t0 , . . . , tn ; c) ∈ Rn+1 >0 × C(M, X) | t0 < · · · < tn , c ∩ S(ti ) = ∅ ∀i .

Let di : Bn → Bn−1 be the map that forgets ti , and ϵ : B0 → C(M, X) be the map that forgets t0 ; with this structure B• is a semi-simplicial topological space, augmented over C(M, X). Proposition 2.1. The map |ϵ| : |B• | → C(M, X) is a weak homotopy equivalence.

Proof. The augmented semi-simplicial space ϵ : B• → C(M, X) is a “topological flag complex” in the sense of [3, Definition 6.1]. Furthermore, it satisfies the conditions of [3, Theorem 6.2]: conditions i) and ii) are clear, and for condition iii) we observe that for any configuration c, and any finite collection of elements of the set {t ∈ R>0 | c ∩ S(t) = ∅}, which is the set of vertices over the configuration c, there exists another element of that set which is larger than them all, as the set is infinite (because the configuration c is finite). Theorem 6.2 of [3] then implies that the augmentation map induces a weak homotopy equivalence on geometric realisation. The space C((0, 1) × S d−1 , X) is an H-space via stacking cylinders ˚ , X) ˚ for M \ e(D1 ) then the spaces C(M end-to-end. If we write M d d−1 and C(R , X) are right and left H-modules over C((0, 1) × S , X) respectively. Thus, fixing a field F,

Configuration spaces of spheres

(i) A := H∗ (C((0, 1) × S d−1 , X); F) is a ring, ˚ , X); F) is a right A-module, (ii) H∗ (C(M (iii) D := H∗ (C(Rd , X); F) is a left A-module. Proposition 2.2. There is a spectral sequence (2.1)

2 ˚ , X); F), D) =⇒ H∗ (C(M, X); F). := TorsA (H∗ (C(M Es,∗

Proof. Let us write Dt ⊂ Rd for the open ball of radius t, and Dt for its closure. There is a map ˚ , X) × (C((0, 1) × S d−1 , X))n × C(Rd , X) ϕ : Bn −→ C(M given by the canonical identifications of ˚, (i) M \ e(Dtn ) with M (ii) Dti+1 \ Dti with (0, 1) × S d−1 , (iii) Dt0 with Rd . The product of ϕ with the map to Rn+1 >0 given by (t0 , . . . , tn ; c) &→ (t0 , t1 − t0 , t2 − t1 , . . . , tn − tn−1 ) is a homeomorphism, and so ϕ is a homotopy equivalence. Furthermore, it is clear that ϕ gives an identification of semi-simplicial objects in the homotopy category of spaces, where ˚ , X) × (C((0, 1) × S d−1 , X))• × C(Rd , X) C(M is such a semi-simplicial object via the H-space and H-module structure maps (and the simplicial identities hold by the homotopy associativity of these maps). If we filter |B• | by skeleta |B• |(k) , then we have identifications |B• |(k) /|B• |(k−1) ∼ = ∆k × Bk /∂∆k × Bk ∼ = S k ∧ (Bk )+ . The spectral sequence for this filtration has 1 = Hs+t (|B• |(s) , |B• |(s−1) ; F) ∼ Es,t = Ht (Bs ; F)

and, following [9, §5], we see that!under this identification the diﬀerential s 1 → E1 i d1 : Es,t s−1,t is given by i=0 (−1) (di )∗ , the alternating sum of

O. Randal-Williams

the maps induced on homology by the face maps. By the identification ϕ and the K¨ unneth theorem we have ˚ , X); F) ⊗ H∗ (C((0, 1) × S d−1 , X); F)⊗s H∗ (Bs ; F) ∼ = H∗ (C(M ⊗ H∗ (C(Rd , X); F) ∼ ˚ , X); F) ⊗ A⊗s ⊗ D = H∗ (C(M 1 , d1 ) and by inspection of the diﬀerential d1 , the chain complex (E∗,∗ ˚ , X); F), A, D). Thus the E 2 agrees with the bar complex B(H∗ (C(M page has the description claimed.

Everything in sight has an extra grading: for ! any manifold N , there is a canonical decomposition H∗ (C(N, X)) = n≥0 H∗ (Cn (N, X)). We call this the multiplicity grading, and write the grading of an element as (h, m) where h is the homological grading and m is the multiplicity grading. Remark 2.3. The notion of topological chiral homology, c.f. [4, §5.3.2], [1], [8], roughly speaking associates to a framed "En -algebra A (in topological" spaces) and an n-manifold N a space N A. The association N $→ N A is covariant, and sends disjoint union to cartesian " product. In particular, for an n-manifold N with boundary the space [0,1]×∂N A there is an A∞ -algebra,#as for each configuration of m little 1-cubes " is an embedding m [0, 1] × ∂N → [0, 1] × ∂N to which − A can be applied. " " It can be shown that N A is a [0,1]×∂N A-module (right or left, as " ′ [0,1]×∂N A has a canonical antiinvolution). Furthermore, if ∂N = ∂N then there is a natural equivalence $% & & $% % L ! A ⊗ A −→ A, A N

[0,1]×∂N

N′

N ∪∂N N ′

from the (derived) tensor product of these two A∞ -modules. This gives a bar spectral sequence $ $% $% $% & && & s ! A , H∗ A =⇒ H∗ A . TorH∗ ( A) H∗ [0,1]×∂N

N′

N ∪∂N N ′

If we take A = C(Rd , X) to be "the free Ed -algebra on a space X, then the topological chiral homology N A may be shown to be homotopy

Configuration spaces of spheres

equivalent to C(N, X), so taking N = M \ int(Dd ) and N ′ = Dd we obtain a spectral sequence " ! ˚ , X)), H∗ (C(Rd , X)) ⇒ H∗ (C(M, X)) TorsH∗ (C([0,1]×∂S d−1 ,X)) H∗ (C(M which agrees with ours.

The structure of A and D in characteristic zero

Let X = ∗ and F = Q. In this section we wish to give a generators and relations description of the ring A and the left A-module D, and construct an explicit resolution (which will have length 1) of D as an A-module. For a smooth manifold with boundary M , a choice of boundary component E gives a stabilisation map sE : Cn (M ) −→ Cn+1 (M ). Let τ M + denote the fibrewise one-point compactification of the tangent bundle of M , and Γn (M ) denote the space of sections of this bundle which are compactly supported in the interior of M , and which have degree n. There is an “electric charge”, or “scanning”, map S : Cn (M ) −→ Γn (M ), cf. [5]. We shall need the following result. We state it for integral homology, though we only need it for rational homology. Proposition 3.1. The map S induces an injection on integral homology, and an isomorphism on integral homology in degrees 2∗ ≤ n. Proof. This is obtained by combining the main results of [5] and [6]. Remark 3.2. In the following, for a set S we write Q[S] for the free commutative Q-algebra on the set S, Q⟨S⟩ for the free noncommutative Q-algebra on the set S, and Q{S} for the free Q-vector space on the set S.

3.1

O. Randal-Williams

The disc: C(Rd )

We write [n] ∈ H0 (Cn (Rd ); Q) for the class of any configuration of n points; these satisfy sE ∗ ([n]) = [n + 1], and [n] has bidegree (0, n). We also write τ ∈ Hd−1 (C2 (Rd ); Q) for the image of the fundamental class under the map S d−1 −→ C2 (Rd )

(3.1)

which sends x to the configuration {0, x}, which has bidegree (d − 1, 2). Proposition 3.3. The class τ 2 ∈ H2(d−1) (C4 (Rd ); Q) is zero, τ and [1] commute, and the induced map ! Q[[1]] d odd φ: −→ H∗ (C(Rd )) = D 2 Q[[1], τ ]/(τ ) d even is an isomorphism. Proof. The scanning map in this case is S : Cn (Rd ) −→ Ωdn S d . By a theorem of Serre, Ωdn S d has trivial rational homotopy groups if d is odd, so also has trivial rational homology, and has a single nontrivial rational homotopy group πd−1 (Ωdn S d ) ⊗ Q ∼ = Q if d is even. It is a simple calculation that in this case it also has a single nontrivial rational homology group in degree (d − 1), and we claim that as long as n ≥ 2 the class S∗ (τ · [n − 2]) is a generator. By the homotopy commutativity of the diagram C2 (Rd ) "

s◦n−2 E

Cn (Rd )

! Ωd S d 2 ≃ −·S([n−2])

" ! Ωd S d , n

and the injectivity of S∗ , it suﬃces to prove that τ ∈ Hd−1 (C2 (Rd ); Q) is nontrivial. But C2 (Rd ) is homeomorphic to RPd−1 , an orientable manifold, and the map (3.1) has degree ±2, so τ is nothing but (±) twice the fundamental class of RPd−1 , hence nontrivial. (For d odd, Hd−1 (RPd−1 ; Q) = 0 so the class τ is zero.) It is clear that τ and [1] commute, by geometric considerations (the multiplication on C(Rd ) extends to an Ed -algebra structure). The class

Configuration spaces of spheres

S∗ (τ 2 ) lies in a group which is zero (as Ωd4 S d has trivial rational homology in degree 2(d − 1), by the above discussion), and S∗ is injective so τ 2 = 0 and we obtain an induced map φ as in the statement of the proposition. This map is clearly an isomorphism in multiplicity grading 0 or 1, and it remains to show that φn : Q{[n], τ · [n − 2]} → H∗ (Cn (Rd ); Q) is an isomorphism for n ≥ 2. But S∗ ◦ φn is an isomorphism, and S∗ is injective, so φn is an isomorphism too, as required. It is convenient to note, as we did in the proof, that the class τ is defined for all d, but is zero if d is odd.

3.2

The cylinder: C((0, 1) × S d−1 )

We perform an analysis similar to the above. There is a map S d−1 −→ C1 ((0, 1) × S d−1 )

(3.2)

sending x to the one-point configuration ( 12 , x), and we let ∆ ∈ Hd−1 (C1 ((0, 1) × S d−1 ); Q) be the image of the fundamental class, so deg(∆) = (d − 1, 1). By d−1 , the lower half of the cylinder, we identifying Rd with (0, 1) × D− obtain an inclusion C(Rd ) −→ C((0, 1) × S d−1 ) and we write [n] and τ for the images of the elements defined in the previous section. Proposition 3.4. In the ring A = H∗ (C((0, 1)×S d−1 ); Q) the relations τ is central

[1] · ∆ = ∆ · [1] − τ

hold. The induced map ! Q[[1], ∆] φ: Q⟨[1], τ, ∆⟩/(τ 2 , [[1], ∆] = −τ, τ central) is an isomorphism.

d odd −→ A d even

O. Randal-Williams

Proof. First consider the proposed relation [1] · ∆ = ∆ · [1] − τ in Hd−1 (C2 ((0, 1) × S d−1 ); Q). This follows from the homology between the cycles ∆ · [1] − [1] · ∆ and τ , which may be seen in the diagram above of cycles in the configuration space of two points on the cylinder. The fact that τ is central follows from similar geometric considerations (clearly τ and [1] commute, then one sees from a similar figure that [∆, τ ] is homologous to a cycle which is supported in a disc, but by the previous section H2(d−1) (C3 (Rd ); Q) = 0 so there are no nontrivial homology classes of this dimension supported in a disc). The target of the scanning map in this case is map∂n ([0, 1]×S d−1 , S d ), the space of continuous maps f : [0, 1] × S d−1 → S d which send {0, 1} × S d−1 to the basepoint ∗ ∈ S d , and which have degree n in the sense that the induced map f∗ : Hd ([0, 1] × S d−1 , {0, 1} × S d−1 ; Z) −→ Hd (S d , ∗; Z) sends the relative fundamental class to n times the fundamental class. The collection of all these mapping spaces fit into a fibration sequence (3.3)

Ωd S d −→ map∂ ([0, 1] × S d−1 , S d ) −→ ΩS d ,

where p restricts a map to the interval [0, 1]×{∗}. By taking the adjoint in the middle mapping space, we can express it as Ωmap(S d−1 , S d ). This exhibits the fibration sequence (3.3) as obtained from looping the evaluation fibration Ωd−1 S d −→ map(S d−1 , S d ) −→ S d so it is a principal fibration. Moreover, the evaluation fibration has a section, given by the inclusion of the constant maps, so after looping it splits as a product: thus (3.3) is a trivial fibration. Let us compute the rational homology of the H-space map∂ ([0, 1] × d−1 S , S d ) with its Pontrjagin ring structure. Suppose first that d is odd. Then Ωdn S d has trivial rational homology, and ΩS d = ΩΣ(S d−1 )

Configuration spaces of spheres

so by the Bott–Samelson theorem has rational Pontrjagin ring the free ! ∗ (S d−1 ; Q), i.e. Q[u] where the class u (non-commutative) algebra on H is obtained from the map S d−1 → ΩS d adjoint to the identity map. As p∗ (S∗ (∆)) = u and p and S are H-space maps, it follows that Q[[1]±1 , S∗ (∆)] −→ H∗ (map∂ ([0, 1] × S d−1 , S d ); Q) is an isomorphism of rings. Suppose now that d is even. We again have that H∗ (ΩS d ; Q) ∼ = Q[u] as a ring. We have the path components [n] ∈ H0 (map∂ ([0, 1] × S d−1 , S d ); Q) for n ∈ Z as well as the classes S∗ (∆) and S∗ (τ ), which we will simply call ∆ and τ again to save space, and we obtain an induced map Q⟨[1]±1 , ∆, τ ⟩ −→ H∗ (map∂ ([0, 1] × S d−1 , S d ); Q). (τ 2 , [[1], ∆] = −τ, τ central) It follows from the fact that p∗ (S∗ (∆)) = u and that (3.3) is a fibration sequence of H-spaces which is trivial as a fibration of spaces (so the homology Serre spectral sequence is one of rings, in fact even of Hopf algebras c.f. [2, §5]) that this map is surjective, but by the splitting of (3.3) and counting dimensions it follows that in fact it is an isomorphism. From these two calculations it follows that the induced map φ in the statement of the proposition is injective. To see that it is an isomorphism, note that the cokernel vanishes after stabilisation by [n] (as φ induces an isomorphism after inverting [1]), so it is enough to show that if x, of bidegree (k(d − 1), m), is such that (3.4)

x · [n] = A · ∆k · [n + m − k] + B · ∆k−1 · τ · [n + m − k − 1]

for n ≫ 0, then if m − k < 0 then A and B are zero, and if m − k = 0 then B is zero. We prove this by induction on the multiplicity grading of x: if m = 0 then the class x is in the homology of C0 ((0, 1) × S d−1 ) = ∗, and the claim follows. For the induction step, we use a map t∗ : H∗ (Cn ((0, 1) × S d−1 ); Q) −→ H∗ (Cn−1 ((0, 1) × S d−1 ); Q) constructed as follows. Let π : Cn,1 ((0, 1) × S d−1 ) → Cn ((0, 1) × S d−1 ) denote the n-fold covering space whose total space consists of a configuration of n points in (0, 1) × S d−1 with one distinguished point, and π forgets which point is distinguished. There is a map f : Cn,1 ((0, 1) ×

O. Randal-Williams

S d−1 ) → Cn−1 ((0, 1)×S d−1 ) which removes the distinguished point, and we let t∗ be the composition of the transfer map for the finite covering π followed by f∗ . The construction of t∗ shows that it is a derivation for the H-space multiplication, and it is easy to compute that t∗ (∆) = 0

t∗ (τ ) = 0

t∗ ([1]) = [0].

If m − k < 0 then applying t∗ to (3.4) n times annihilates ∆k · [n + m − k] and ∆k−1 · τ · [n + m − k − 1], and we obtain n! · x + y · [1] = 0 for y some expression in iterated applications of t∗ to x. The class y is of bidegree (k(d − 1), m − 1) and satisfies the analogue of (3.4), so by induction both A and B are zero, which finishes the proof in this case. If m − k = 0 then applying t∗ to (3.4) n times gives n! · x + y · [1] = A · n! · ∆k and substituting back into (3.4) gives y ·[n+1] = −n!·B ·∆k−1 ·τ ·[n−1]. But y has bidegree (k(d−1), m−1) so by induction B = 0, which finishes the proof in this case.

3.3

D as an A-module (d even)

The left A-module structure on D is given by [k] • (τ ϵ · [n]) = τ ϵ · [n + k]

τ • (τ ϵ · [n]) = τ ϵ+1 · [n]

and ∆ • (τ ϵ · [n]) = n · τ ϵ+1 · [n − 1]. The first two are clear and the last follows from ∆ • [0] = 0 (as C1 (Rd ) is contractible, so has trivial homology in degree (d − 1)), ∆ • τ = 0 (as C3 (Rd ) has trivial homology in dimension 2(d − 1)), and the commutation relation [∆, [1]] = τ . One verifies that 0

! Σd−1,1 A

·∆

is an exact sequence of left A-modules.

[0]#→[0]

Configuration spaces of spheres

3.4

D as an A-module (d odd)

The left A-module structure on D is given by [k] • [n] = [n + k] and ∆ • [n] = 0. One verifies that 0

! Σd−1,1 A

·∆

! A [0]"→[0]! D

is an exact sequence of left A-modules.

Configuration spaces of spheres

We apply the spectral sequence (2.1) to M = S d , for d even. In this case ˚ = Dd so H∗ (C(M ˚ ); Q) = D, but it has a right A-module structure. M This is induced from the left A-module structure by the antiinvolution of A, which in turn is induced by reflecting the first coordinate of the cylinder around 12 . It is easy to see that this antiinvolution : A → A is given on generators by [n] = [n]

τ = −τ

∆ = ∆.

Concretely, the right module structure is given by (−) • (∆i · τ ϵ · [n]) := ([n] · (−τ )ϵ · ∆i ) • (−). We have shown that D has a length 1 resolution by free A-modules, −•∆ so the complex Σd−1,1 D −→ D computes the E 2 page of the spectral sequence. This map is given explicitly by (τ ϵ · [n]) #→ (τ ϵ · [n]) • ∆ = ∆ • (τ ϵ · [n]) = n · τ ϵ+1 · [n − 1], so is as shown in Figure 1. By inspection there can be no further diﬀerentials in the spectral sequence, and we immediately see the correct rational homology for C0 (S d ) = ∗, C1 (S d ) = S d , and C2 (S d ) ≃ RP d , and also deduce that for all n ≥ 3, Cn (S d ) has the rational homology of S 2d−1 . The analysis for d odd is the same, but a little easier as the element τ does not appear.

Final remarks

The reader will realise that the multiplicative decomposition technique described in Section 2 admits many variations. Let us describe one,

O. Randal-Williams

Figure 1: E 1 page of the spectral sequence for d even, with multiplicity grading along the horizontal axis, and Tor grading along the vertical axis. The homological grading is not shown. which leads to what appears to be a diﬃcult calculation in homological algebra. For the manifold M = S 1 × S d−1 we may let Bn be " ! # " p distinct, cyclically ordered, 1 n+1 (p0 , . . . , pn ; c) ∈ (S ) × C(M ) "" i c ∩ {pi } × S d−1 = ∅

and as in Section 2 we may show that the augmentation |B• | → C(M ) is a homotopy equivalence. Following the proof of Proposition 2.2, we find that the E 1 page of the resulting spectral sequence is now the cyclic bar complex for the algebra A = H∗ (C((0, 1) × S d−1 ); F), so we have a spectral sequence 2 Es,∗ = HHs (A, A) =⇒ H∗ (C(S 1 × S d−1 ); F).

starting with the Hochschild homology of the algebra A with coeﬃcients in itself. For F = Q, one ought to be able to use the calculation in Section 3.2 of the algebra A to study this spectral sequence, but the homological algebra seems to be a lot harder. Oscar Randal-Williams Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK, o.randal-williams@dpmms.cam.ac.uk

References [1] Andrade R., From manifolds to invariants of En -algebras, arXiv:1210.7909 [math.AT],(30 Oct 2012).

Configuration spaces of spheres

[2] Browder W., On diﬀerential Hopf algebras, Trans. Amer. Math. Soc. Transactions of the American Mathematical Society 107 1963, 153–176. [3] Galatius S.; Randal-Williams O., Homological stability for moduli spaces of high dimensional manifolds, arXiv:1203.6830 [math.AT], (30 Mar 2012). [4] Lurie J., Higher Algebra, 2012. [5] McDuﬀ D., Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. [6] Randal-Williams O., Homological stability for unordered conguration spaces, Q. J. Math 64:1 (2013), 303–326. [7] Salvatore P., Configuration spaces on the sphere and higher loop spaces, Cohomological Methods in Homotopy Theory, Barcelona Conference on Algebraic Topology, Bellaterra, Spain (1998), Progress in Mathematics, 196 2001, 375–395. [8] Salvatore P., Conguration spaces with summable labels, Q. J. Math 64:1 (2013), 303–326. [9] Segal G., Classifying spaces and spectral sequences, Inst. Hautes ´ Etudes Sci. Publ. Math. 34:1 (1968), 105–112. [10] Weiss M., What does the classifying space of a category classify?, Homology Homotopy Appl. 7:1 (2005), 185–195.

Morfismos, Vol. 17, No. 2, 2013, pp. 71–100

Cooperads as symmetric sequences Benjamin Walter

Abstract We give a brief overview of the basics of cooperad theory using a new definition which lends itself to easy example creation and verification and avoids common pitfalls and complications caused by nonassociativity of the composition operation for cooperads. We also apply our definition to build the parenthesization and cosimplicial structures exhibited by cooperads and give examples.

2010 Mathematics Subject Classification: 18D50; 16T15, 17B62. Keywords and phrases: Cooperads, operads, coalgebras, Kan extensions.

Introduction

In the current work we discuss cooperads in generic symmetric monoidal categories from the point of view of symmetric sequences. Fix a symmetric monoidal category (C, ⊗). Let us roughly recall the standard framework. Operads encode algebra structures. The tautological example is the endomorphism operad of an object end (A) = n Hom(A⊗n , A). Operads have a natural grading by levels expressing the “arity” of diﬀerent “operations” (for example, end (A)(n) = Hom(A⊗n , A)). The symmetric group Σn acts on the n-ary operations of an operad (for end (A)(n) this action is by permutation of the A⊗n ). A graded object with Σn actions is called a “symmetric sequence.” Operads are further equipped with a composition product identifying the result of plugging operations into each other (for example, end (A) ◦ end (A) → end (A)). Very roughly, an operad is “a bunch of objects with a rule for plugging them into each other”. Operads encode algebra structures via maps of operads (preserving symmetric group actions and composition structure). So, for example, 71

B. Walter

there is an operad lie of formal Lie bracket expressions modulo Lie relations, along with a composition rule identifying the result of plugging bracket expressions into each other. A map of operads lie → end(A) identifies a specific endomorphism of A for each formal Lie bracket expression, in such a way that composition of Lie bracket expressions is compatible with the composition of corresponding endomorphisms. This gives A the structure of a Lie algebra. Coalgebra structures can also be defined The coendo! via operads. ⊗n morphisms of an object coend(A) = n Hom(A, A ) also form an operad: It is graded, with symmetric group action, and has a natural map coend(A) ◦ coend(A) → coend(A) also given by plugging things into each other. Replacing end by coend changes algebra structures to coalgebra structures. For example a map of operads lie → coend(A) identifies a coendomorphism of A for each Lie bracket expression, thus giving A a Lie coalgebra structure. This is a common point of view (see e.g. [11]), but there is an alternative. For clarity, we will continue with the example of Lie algebras. A Lie algebra structure is maps lie(n) → Hom(A⊗n , A) which is equivalent to maps lie(n) ⊗ A⊗n → A (ignore Σn -actions for the moment). Dually, a Lie coalgebra structure is maps lie(n) → Hom(A, A⊗n ) which " is equiva#∗ lent to maps lie(n) ⊗ A → A⊗n which is equivalent to A → lie(n) ⊗ A⊗n . (Dualizing lie(n) should not introduce trouble, #∗ it is finite ! " because dimensional.) The level-wise dual object lieˇ= n lie(n) has structure dual to that of lie. This is a cooperad. (The precise definition is the subject of the current paper.) Experience [9] [10] has shown that it is sometimes more useful to directly work with cooperads and cooperad structures when describing coalgebras rather than continually referring all the way back to operads and operad structures. Also sometimes coalgebras can have a more natural expression as coalgebras over cooperads, rather than coalgebras over operads. Just as operads can be thought of as “a bunch of objects which are plugged into each other”, cooperads can be thought of as “a bunch of objects where subobjects are removed or quotiented”. Unfortunately category theory causes a slight hitch when attempting to blindly dualize operad structure to define cooperads. The dual of operad composition is cooperad composition, which is similar except for some colimits being replaced by limits. The problem comes when looking at associativity. In a symmetric monoidal category ⊗ is left adjoint (to Hom) so it will commute with colimits. This allows operad composition products to be associative (e.g. (lie◦lie)◦lie = lie◦(lie◦

Cooperads as Symmetric Sequences

lie)). However, this will generally not happen for cooperad composition (e.g. (lieˇ• lieˇ) • lieˇ ̸= lieˇ• (lieˇ• lieˇ)). This issue crops up for example, in the cooperadic cobar constructions of Ching in his thesis [4] and arXiv note [5]. We work by defining a new composition product – a composition product of tree-functors. The motivating intuition is that the composition product of two symmetric sequences should not itself be a symmetric sequence – in particular its group of symmetries is much too large. Maps to and from the tree-functor composition product can be expressed as maps to and from universal extensions, which yields the classical operad and cooperad composition products. Using the tree-functor composition product (rather than its Kan extension) when describing or defining cooperads greatly simplifies bookkeeping; though it turns out that, for operads, it doesn’t really make a diﬀerence. We begin by introducing the notation of wreath product categories. These are inspired by the wreath product categories of Berger [2], and at the most basic level are merely Groethendieck constructions. Wreath product categories are defined so that they will be the natural source category for iterated composition products of symmetric sequences. We use this to give a simple definition of cooperads and prove all of the standard structure holds. Then we describe comodules and coalgebras. We finish with simple examples related to work in [9], [10], and [13]. In the sequel [12] we use the structure presented here to build cofree coalgebras, connecting to the constructions of Fox [6] and Smith [11]. We assume that the reader is comfortable with the category theory notions of adjoint functors and Kan extensions, as well as basic simplicial and cosimplicial structures. A familiarity with the classical definitions of operads and their modules/algebras is not required, but would be helpful.

Wreath product categories

This section is divided into two parts. In the first subsection, we define wreath product categories using functors to the category of finite sets. Our definition is related to, but more general than, the dual of refined partitions of sets as used in literature by e.g. Arone-Mahowald [1]. The salient diﬀerence between wreath categories and refined partitions is that wreath categories incorporate the empty-set (see Remark 2.8). In the second subsection, an equivalent definition is given in terms of

B. Walter

labeled level trees – a more familiar category for the discussion of operads. The neophyte reader (or the reader looking for an immediate connection to classical constructions) may find it useful to read §2.2 before §2.1.

2.1

Wreath Products

Write ! Σn for the category of n-element sets and set isomorphisms and Σ∗ = n≥0 Σn for the category of all finite sets and set isomorphisms (Σ0 = ∅). Our notation reflects the fact that a functor Σn → C is merely an object of C with a Σn -action. There is an alternative way to express Σn . Write FinSet for the category of finite sets and all set maps, and write [n] for the category f1

fn−1

1 −→ 2 −→ · · · −−−→ n. Then Σ∗ is equivalent to the category of functors [1] → FinSet and natural isomorphisms. We generalize this to define wreath product categories. Definition 2.1. The wreath product category Σ≀n ∗ is the category of contravariant functors [n] → FinSet and natural isomorphisms. Remark 2.2. We will write objects of Σ≀n ∗ as sequences of maps of finite sets, indexed in the following manner. f1

fn−1

S1 ←− S2 ←− · · · ←−−− Sn op Since [n] ∼ = [n] , the use of contravariant functors in Definition 2.1 is purely cosmetic. Using covariant functors would change nothing, except that indices would not line up as perfectly later on.

Note that we are clearly defining the levels of a simplicial category. Before continuing in that direction, however, we will explain our choice of notation via an equivalent, hands-on definition of wreath products with a generic category A. Definition 2.3. The wreath product category Σn ≀ A is the category with # " $ • objects Obj(Σn ≀ A) = {As }s∈S # S ∈ Obj(Σn ), As ∈ Obj(A) given by n-element sets of decorated objects of A; & % • and morphisms σ; {φt }t∈T : {At }t∈T −→ {Bs }s∈S given by a set isomorphism σ : T → S and a set of A-morphisms φt : At → Bσ(t) .

Cooperads as Symmetric Sequences

The wreath product category Σ∗ ≀A is given by Σ∗ ≀A :=

n≥0 Σn ≀A.

Remark 2.4. Σ0 ≀A is the empty category, since {As }s∈∅ = ∅. Furthermore Σ∗ ≀ Σ0 ∼ = Σ∗ ∼ ="Σ1#≀ Σ∗ . These equivalences are given " # by writing objects of Σ∗ ≀ Σ0 as ∅s s∈S and objects of Σ1 ≀ Σ∗ as S⋆ and using the facts that ∅ is initial and a one point set ⋆ is final in FinSet. We make further use of these equivalences later. Note that Σ∗ ≀ Σ1 ! Σ∗ because one point sets are not initial in FinSet. The following proposition is easy to check. Proposition 2.5. Definitions 2.1 and 2.3 agree: ∼ • Σ≀2 ∗ = Σ∗ ≀ Σ∗ , and more generally n

'( ) & ∼ ∼ • = Σ∗ ≀ = Σ∗ ≀ (· · · ≀ (Σ∗ ≀ Σ∗ )). $ % f Proof Sketch: The object S ← − T of Σ≀2 ∗ corresponds to the object # " −1 f (s)s s∈S of Σ∗ ≀ Σ∗ . * % " # $ π The object As s∈S corresponds to S ←− As where π is the Σ≀n ∗

% Σ∗≀n−1

s∈S

map on the coproduct induced by πs : As → {s} ⊂ S.

Using notation from Definition 2.3, the endomorphisms of the wreath product category Σn ≀ Σm correspond to the automorphisms of an nelement set of m-element sets S = {A1 , . . . , An } with |Ai | = m. Elements within each Ai can be permuted by Σm and the Ai “blocks” are permuted by Σn – this is the wreath product group Σn ≀ Σm . Thus, a functor Σn ≀ Σm → C is an object of C equipped with an action of the wreath product group Σn ≀ Σm . We view Σ∗ ≀ Σ∗ as a generalization of this basic example – the “blocks” Ai no longer need to be same size, and there can be an arbitrary number of them. We return to the simplicial structure of the collection of wreath ! ≀n products n Σ∗ . Recall that there are standard “face” functors ∂in : [n] → [n − 1]

for 1 ≤ i ≤ (n − 1), given by composing morphisms or forgetting 1 (for reasons to be explained shortly, we do not use the “forget n” face map, ∂nn ). % $ % fn−1 f1 ∂1n (1 −→ · · · −−−→ n = 2 → · · · → n $ f1 % $ % fi ◦fi−1 fn−1 ∂in 1 −→ · · · −−−→ n = 1 → · · · → (i − 1) −−−−→ (i + 1) → · · · → n

B. Walter

Furthermore, (because we do not allow the use of ∂nn functors) any chain of (n − 1) compositions ∂i22 ◦ · · · ◦ ∂inn : [n] → [1] equals the functor γ n : [n] → [1] which forgets all but the top object. " ! " ! f1 fn−1 γ n 1 −→ · · · −−−→ n = n

≀(n−1)

We will write ∂in and γ n also for the induced functors ∂in : Σ≀n , ∗ → Σ∗ ≀n n for 1 ≤ i ≤ (n − 1), and γ : Σ∗ → Σ∗ . When n is clear from context we may write merely ∂i and γ. Remark 2.6. In the notation of Definition 2.3, the map γ 2 = ∂12 : # Σ∗ ≀ Σ∗ → Σ∗ is given by {St }t∈T &→ T St . All other ∂in and γ n are induced by this (see Proposition 2.10). Before describing the degeneracy maps, we explain the missing ∂nn . $ ∗ = Σ1 ≀ Σ∗ ⊂ Recall that Σ∗ is equivalent to the full subcategory Σ Σ∗ ≀ Σ∗ of functors sending 1 to a one-element set. More generally, Σ≀n ∗ ≀n ≀n+1 $ ≀n is equivalent to the full subcategory Σ = Σ ≀ Σ ⊂ Σ of functors 1 ∗ ∗ ∗ $ ≀n sending 1 to a one-element set. Objects of Σ ∗ are sequences of set maps f0

fn−1

⋆ ←− S1 ←− · · · ←−−− Sn

$ ≀n $ ≀n−1 , for Under this correspondence the face functors ∂˜in : Σ ∗ → Σ∗ 1 ≤ i ≤ (n − 1), are all given by composition; however the functor ∂˜nn is not. ! f0 " fn−1 f1 ∂˜in ⋆ ←− S1 ←− · · · ←−−− Sn = " ! fi−1 ◦fi ⋆ ← · · · ← Si−1 ←−−−− Si+1 ← · · · ← Sn

$ ≀n (Our indexing convention is for the one point set to be ⋆ = S0 in Σ ∗ ). $ ≀n and Operad and cooperad structure is induced by structure of Σ ∗ ∂˜in . Instead of working with this directly, we use the equivalent catn egories and functors Σ≀n ∗ and ∂i ; because in practice keeping track of the final, one point set at the bottom of each sequence is unnecessarily tedious. We continue with the degeneracies of the simplicial structure, which $ ≀n are most conveniently written via the equivalent categories Σ ∗ . In this ≀n ≀n+1 n $∗ → Σ $∗ for 0 ≤ i ≤ n are notation, the degeneracy functors s˜i : Σ the doubling maps. " ! f0 " ! fn−1 f1 Id s˜ni ⋆ ←− S1 ←− · · · ←−−− Sn = ⋆ ← · · · ← Si ←− Si ← · · · ← Sn

Cooperads as Symmetric Sequences

Note that defining the degeneracy sn0 on the level of Σ≀n ∗ requires picking a distinguished one point set. A reader averse to making choices should ! ∗ , ∂˜i , etc. from now on. replace all Σ∗ , ∂i , etc. by Σ It is classical that the degeneracies sni−1 and sni are each sections of the face map ∂in+1 on the level of [n]. Thus face and degeneracy maps combine to give a collection of categories and functors: Σ∗ ≀ Σ∗ ≀ Σ∗ ≀ Σ∗

···

Σ∗ ≀ Σ∗ ≀ Σ∗

Σ∗ ≀ Σ∗

Σ∗

where the dashed, left-pointing arrows are sections of their neighboring right-pointing arrows and all pairs of neighboring right-pointing arrows are coequalized by an arrow out of their target. Under the correspon≀n+1 ∼ ! ≀n dence Σ≀n , this is very explicitly a simplicial category ∗ = Σ∗ ⊂ Σ∗ with the bottom level as well as the first and last face maps removed; equivalently, an augmented simplicial category with two extra degeneracies. Remark 2.7. We could express all of the standard face maps ∂in , 1 ≤ ≀n ≀n+2 ∼ ≀n , the i ≤ n, as compositions by writing Σ≀n ∗ = Σ∗ = Σ1 ≀ Σ∗ ≀ Σ0 ⊂ Σ∗ full subcategory of functors sending (n + 2) to the empty-set and 1 to a one-element set. Then ∂nn becomes: fn−1 fn # f1 f0 n" ∂ n ⋆ ←− S1 ←− · · · ←−−− Sn ←− ∅ = ≀n

" fn−1 ◦fn # ⋆ ← S1 ← · · · ← Sn−1 ←−−−−− ∅

The Σ∗ fit together to make an (unaugmented) simplicial category with two extra degeneracies. In the next section, the levels of this will be given an alternate definition and called ˆ∅n . This structure is useful for constructing algebras and coalgebras instead of operads and cooperads. Remark 2.8. Another construction which has been useful in the past for describing and working with operads uses the category of sets equipped with iterated refinements of partitions where morphisms are given by set isomorphisms respecting all partition equivalences (see AroneMahowald [1] and Ching [4]) . A partition of a set S is equivalent to a surjective set map S → T where T is the partition set. An iterated partition of a set S is equivalent to a functor from [n] to the category of finite sets and surjections (instead of the category of finite sets and all set maps). This is suﬃcient for describing operads and cooperads

B. Walter

which are trivial in “0-arity”. So partitions cannot be used to describe, for example, an operad of algebras over an algebra. Also missing 0-arity means that partitions cannot work with algebras (or coalgebras) as just a special case of modules (or comodules). Before continuing with the next subsection, we will combine Definitions 2.1 and 2.3 to get a more general definition of wreath products with generic categories, necessary to discuss associativity. Definition 2.9. The wreath product category Σ≀n ∗ ≀ A is the category with !" $ # ≀n • Obj(Σ≀n | F ∈ Obj(Σ ≀ A) = F, {A } ), A ∈ Obj(A) s s∈F (n) s ∗ ∗

# " # " # " • morphisms Φ; {φs }s∈F (n) : F, {As } → G, {Bt } given by a natural isomorphism Φ : F → G and a set of A-morphisms φs : As → B(Φn)(s) Objects of Σ≀n ∗ ≀ A can be written as sequences of set maps & fn−1 f1 f2 fn % S1 ←− S2 ←− · · · ←−−− Sn−1 ←− As s∈Sn .

Morphisms are level-wise set isomorphisms accompanied by (at the top ∼ = level) A-maps As → Bt (where φ : Sn −→ Tn with φ(s) = t). In the following subsection we will give an alternate way to describe these objects via labeled trees. Proposition 2.10. Wreath product is associative: (Σ∗ ≀ Σ∗ ) ≀ Σ∗ ∼ = Σ∗ ≀ (Σ∗ ≀ Σ∗ ) ∼ = Σ≀3 ∗. ≀m ∼ ≀n+m . Furthermore, the face maps ∂in are More generally, Σ≀n ∗ ≀ Σ∗ = Σ∗ all induced by γ 2 = ∂12 as # " # " ∂in = Id ≀ γ 2 ≀ Id : Σ∗≀i−1 ≀ Σ∗ ≀ Σ∗ ≀ Σ∗≀n−i−1 −→ Σ∗≀i−1 ≀ Σ∗ ≀ Σ∗≀n−i−1 .

For example ∂13 = γ 2 ≀ Id and ∂23 = Id ≀ γ 2 .

2.2

Level trees

In this subsection we connect the wreath product constructions of the previous subsection with the classical, visual, method of describing operads via trees.

Cooperads as Symmetric Sequences

For our purposes a tree is a (nonempty) non-cyclic, connected, finite graph whose vertices are distinguished as: a “root vertex” of valency 1, a (possibly empty) set of “leaf vertices” of valency 1, and all other vertices called “interior vertices”. We require each tree to have a root and at least one interior vertex; however, we do not require that interior vertices have valency > 1 – despite the oxymoron (in particular, we allow the tree with a root, an “interior vertex” but no leaves as in Figure 1). A tree isomorphism is an isomorphism of vertex and edge sets, preserving root and leaf distinctions. For convenience of notation we will orient all edges of our trees so that they point towards the root vertex; when drawing trees, we will not explicitly indicate this orientation, but rather always position the root at the bottom and the leaves at the top, with the understanding that all edges point downwards. We will denote interior vertices with a darkened dot •, but we will not draw the root or leaf vertices – instead we will indicate only the edges connecting to them. Also for convenience, we will draw trees on the plane, however we consider them as non-planar objects. In particular, we will not assert any planar orderings on vertices or edges. There is a natural height function on the vertices of trees – assigning to each vertex the number of vertices on the path between it and the root (the vertex adjacent to the root has height 0; the root has height -1). A “level n tree” is a tree whose leaves all have height n and whose interior vertices have height < n. A “level tree” is a tree which is level n for some n. Note that a level n tree may have branches without leaves which contain no interior vertices of height (n − 1), as in Figure 1. In particular, a tree with no leaves may be level n as well as level (n + 1), etc.

•

• • •

• •

• • •

•

• •

•

• •

• • •

• •

• • •

•

Figure 1: Some examples of level 2 trees and a level 4 tree If v is the target of the directed edge e then we say e is an “incoming edge” of v and we write In(v) for the set of incoming edges of v. In our drawings, incoming edges are edges abutting a vertex from above. Each

B. Walter

non-root vertex also has one “outgoing edge” (the abutting edge on the path from the vertex to the root), which will be drawn connecting to the vertex from below. Definition 2.11. A labeled level tree is a level tree equipped with ∼ = labeling isomorphisms {lv : Sv −−→ In(v)}v from finite sets to the sets of incoming edges at each vertex. Let Ψ be the category of all labeled level trees with morphisms given by tree isomorphisms. Let Ψn be the full subcategory of Ψ consisting of only level n trees. Since there is always only one incoming edge at the root, and never any incoming edges at leaves, we may equivalently label only the incoming edges at interior vertices. Definition 2.12. Given a category A define the wreath product category Ψ ≀ A to be the category of all labeled level trees whose leaves are decorated by elements of A; morphisms are given by tree isomorphisms equipped with A-morphisms between the leaf decorations compatible with the induced isomorphism of leaf sets. Let Ψn ≀ A be the full subcategory of this consisting of only level n trees. i

•

i j k

•

Ai i

Ai Aj i j

Ai Aj Ak i j k

•

Figure 2: Some objects of Ψ1 and of Ψ1 ≀ A It is standard to note that the category Σ∗ may be identified with the category Ψ1 of labeled level 1 trees. In this vein, the wreath product category Σ∗ ≀A may be identified with Ψ1 ≀A. More generally, the wreath product category Σ∗ ≀ Σ∗ is equivalent to the category Ψ2 of all labeled level 2 trees; and the iterated wreath product category Σ≀n ∗ is equivalent to Ψn the category of all labeled level n trees. Proposition 2.13. There are equivalences of categories: Ψ1 ∼ = Σ∗ ,

Ψ1 ≀ A ∼ = Σ∗ ≀ A,

≀n Ψn ∼ = Σ∗ ,

and

≀n Ψn ≀ A ∼ = Σ∗ ≀ A.

! ≀2 Example 2.14. The elements of Σ ∗ corresponding to the Ψ2 elements in Figure 3 are given by the following chains of maps in FinSet: " # • ⋆←∅←∅ .

Cooperads as Symmetric Sequences

j1 j2

•

j1 j2 j3

k1 k2

• •

•

• •

•

i1 i2

•

Figure 3: Some objects of Ψ2 ! " • ⋆ ← {i2 , i2 } ← ∅ . ! " f • ⋆ ← {i} ← − {j} where f (j) = i. " ! f − {j1 , j2 } where f (js ) = i1 . • ⋆ ← {i1 , i2 } ← ! " f • ⋆ ← {i1 , i2 , i3 } ← − {j1 , j2 , j3 , k1 , k2 } where f (js ) = i1 while f (kt ) = i3 .

Under this identification, the functor γ 2 = ∂12 : Ψ2 → Ψ1 operates by forgetting the height 1 vertices on a level 2 tree. Paths from the height 0 interior vertex to leaves (on level 2) are replaced by edges; the labeling of each such edge is given by the path labeling of the path which it replaces, as in Figure 4. 2

γ =

∂12

j1 j2 j3

k1 k2

•

i1 j 1 i1 j 2 i1 j 3 i3 k 1 i3 k 2

%−→

•

Figure 4: An example of γ 2 : Ψ2 → Ψ1 Similarly, the functors γ n : Ψn → Ψ1 operate by forgetting all interior vertices except for those of height 0; replacing paths by edges carrying the paths’ labels. The face functors ∂in : Ψn → Ψn−1 for 1 ≤ i ≤ n−1 are given by forgetting only the vertices of level i of a level n tree. (The disallowed face functor ∂nn would forget the leaves.) The degeneracy functors sni : Ψn → Ψn+1 for 0 ≤ i ≤ n are given by “doubling” – replace each vertex v at level i by two vertices connected by a directed edge ev , attached to the tree such that all incoming edges connect to source vertex of ev and the outgoing edge connects to the target vertex (for the labeling, allow each edge to label itself lt(ev ) : {ev } → {ev }). Note that the degeneracy snn doubles the leaf vertices – the leaves of the resulting tree are the sources of the edges ev . Remark 2.15. We very purposefully do not use the notation Υ for our category of level trees, since that notation is already commonly used

B. Walter

s10 :

i j k

•

!−→

i j k

j k

•

s11 :

j k

•

!−→

• • • e1 e2 e3

•

• Figure 5: An example of s10 , s11 : Ψ1 → Ψ2

to denote the category consisting of all trees. The category Ψ diﬀers from this both on the level of objects (only level trees) and on the level of morphisms (only isomorphisms of trees – in particular, no “edge contraction” maps). Remark 2.16. Note that Ψ is not isomorphic to the category Ψ∗ = ! ˆ n Ψn . Write ∅n for the full subcategory of Ψn consisting of trees with no leaves. Then ˆ ∅n is a full subcategory of ˆ∅n+1 . In terms of the Ψn , the category Ψ itself is given by " " " Ψ∼ Ψ2 Ψ3 Ψ4 · · · = Ψ1 ˆ ∅1

ˆ ∅2

ˆ ∅3

In the notation of the previous subsection, an element of ˆ∅n is equivalent to a contravariant functor [n] → FinSet sending n to the empty-set as in Remark 2.7.

3 3.1

Symmetric sequences, composition products, and cooperads Symmetric Sequences

Let (C, ⊗, 1⊗) be a symmetric monoidal category with monoidal unit 1⊗. In order to have all desired Kan extensions exist, we will further require that C is cocomplete. Write ⋆C for the final object of C. [In order to dualize to operads, we would require C be complete with initial object ∅C .] Definition 3.1. A symmetric sequence is a functor A : Σ∗ → C. Recall that a functor Σ∗ → C is equivalent to a sequence of objects {A(n)}n≥0 of C along with a symmetric group action on each A(n). We will make use of this viewpoint when convenient without further

Cooperads as Symmetric Sequences

comment. If A is a symmetric sequence, then we will refer to A(n) as the “n-ary part of A” since for operads it will encode n-ary algebra operations. (The “0-ary operations” require no input. For example, in the category of algebras over a field, elements of the base field are all 0-ary operations.)

3.2

Composition of Symmetric Sequences

We define a “product” operation on symmetric sequences. It is important to note that our product will not itself be a symmetric sequence. Instead it is a larger diagram, reflecting a larger group of symmetries. The traditional composition product of operads as well as our cooperad composition product are Kan extensions of this symmetric sequence product. Definition 3.2. Given A1 , . . . , An : Σ∗ → C define (A1 ! · · · ! An ) : ! ≀n Σ ∗ → C by ⎛ ⎞ $ $ " f0 # " −1 # fn−1 f1 ⎝ ⋆ ←− S1 ←− · · · ←−−− Sn $−→ Ai+1 fi (s) ⎠ 0≤i≤n−1

s∈Si

with the convention that ⋆ = S0 . Define A1 • · · · • An to be the right Kan extension of A1 ! · · · ! An over the map γ : Σ≀n ∗ −→ Σ∗ . Σ∗ γ

A1 •···•An := Rγ A1 !···!An ι

Σ∗ ≀ · · · ≀ Σ∗

A1 !···!An

Write ι : (A1 • · · · • An ) γ → A1 ! · · · ! An for the universal natural transformation. [Dually, to construct operads , we would define A1 ◦ · · · ◦ An to be the left Kan extension over γ.] Remark 3.3 (For young readers). The symmetric sequence A • B is completely determined by the property that every natural transformaζ

tion Cγ − → A ! B factors uniquely as Cγ − → (A • B)γ − → A ! B. Dually, A ◦ B is determined by the the unique factorization of every ρ

→ Dγ as A ! B −→ (A ◦ B)γ −→ Dγ. transformation A ! B −

B. Walter

Using the notation of Definition 2.9, we can generalize the above definition slightly in order to discuss associativity. Definition 3.4. Given A : Σ≀n ∗ → C and B : A → C, define (A ! B) : ≀n Σ∗ ≀ A −→ C by ⎞ ⎛ $ # ' ! " B(As )⎠ . A ! B F, {As }s∈F (n) = A(F ) ⊗ ⎝ s∈F (n)

Remark 3.5 (Completed tensor product). The discussion of coalgebras via cooperads in the introduction glossed over an important subtlety. The dual of the endomorphism operad is * * ˆ Hom(A∗ , (A⊗n )∗ ) = Hom(A∗ , (A∗ )⊗n ) n

ˆ is the “completed” tensor product (if C ∗ is a category where where ⊗ ˆ is a right adjoint (rather than left adjoint) this exists). The product ⊗ ˆ is usually something like “formal, to Hom. In cases where it exists, ⊗ infinite linear combinations of elements a ⊗ b”. For coalgebras to satisfy a maximal number of duality properties with algebras, the completed tensor product (if it exists) should make an appearance once we begin discussing coalgebras; but it is not used in the construction of cooperads. Note that in categories satisfying good finiteness conditions (for example finitely generated projective bimodules over a commutative ring), A∗ ⊗ ˆ B ∗. B ∗ = (A ⊗ B)∗ = A∗ ⊗ Short calculations yield the following propositions. Proposition 3.6. The operation ! is associative: (A1 ! A2 ) ! A3 ∼ = A1 ! A2 ! A3 ∼ = A1 ! (A2 ! A3 ). Proposition 3.7. Given A, B symmetric sequences, A • B is given by ⎞ ⎛ Σk + + ⎝ A(k) ⊗ B(r1 ) ⊗ · · · ⊗ B(rk )⎠ . (A • B)(n) = k≥0

ri = n

Note that • is probably not associative. This will be discussed in greater detail in the next section (see Proposition 4.1). The operation ! is clearly functorial. If F : A1 → A2 and G : B1 → B2 are natural

Cooperads as Symmetric Sequences

≀m transformations of functors A1 , A2 : Σ≀n ∗ → C and B1 , B2 : Σ∗ → C, then we write (F ! G) : (A1 ! B1 ) → (A2 ! B2 ) for the induced natural ≀m transformation of functors Σ≀n ∗ ≀ Σ∗ → C.

In the following subsections, we define cooperad structure and, in ! parallel, build the cosimplicial structure induced on n A!n by the ! simplicial structure of wreath product categories n Σ≀n ∗ .

3.3

Cocomposition and Coface Maps

˜ Definition 3.8. A symmetric sequence with cocomposition is (A, ∆) 2 ˜ ˜ where ∆ is a cocomposition natural transformation ∆ : A γ −→ A ! A of functors Σ∗ ≀ Σ∗ → C compatible with the face maps ∂13 = (γ 2 ≀ Id) and ∂23 = (Id ≀ γ 2 ). Write ∆ for the associated universal natural transformation of symmetric sequences ∆ : A −→ A • A. In other words, the following diagram of functors Σ∗ ≀ Σ∗ ≀ Σ∗ → C should commute. ˜ ∆

(1)

(A ! A)(γ 2 ≀ Id)

Aγ 3

˜ ∆≀Id

A!A!A ˜ ∆

(A ! A)(Id ≀ γ 2 )

˜ Id≀∆

The upper path uses the factorization γ 3 = ∂12 ◦ ∂13 = γ 2 ◦ (γ 2 ≀ Id) and the lower path uses the factorization γ 3 = ∂12 ◦ ∂23 = γ 2 ◦ (Id ≀ γ 2 ). ˜ to the following Applying Proposition 2.10, we may generalize ∆ maps. Definition 3.9. For a given a symmetric sequence with cocomposition ˜ define associated natural transformations ∆ ˜ n : A!(n−1) ∂ n → (A, ∆) i i !n ˜ at position i. (Thus ∆ ˜ =∆ ˜ 2 .) A , for 1 ≤ i ≤ (n − 1), which apply ∆ 1 ! !n These natural transformations induce coface maps on n A in the following manner. Since γ n−1 ∂in = γ n and ∂in is epi, transformations B γ n → A!(n−1) ∂in are equivalent to transformations Bγ n−1 → A!(n−1) (where B : Σ∗ → C is some symmetric " sequence).# Therefore" there is#an equality of right Kan extensions Rγ n A!(n−1) ∂in = Rγ n−1 A!(n−1) = A•(n−1) . (We will make extensive use of this equality in later sections without further comment.)

B. Walter

Define ∆ni : A•(n−1) → A•n to be the following map. (2)

A•(n−1) ! " Rγ n A!(n−1) ∂in

˜ n) Rγ n (∆ i

A•n ! " Rγ n A!n

Under right Kan extension, Diagram (1) translates to the following diagram of symmetric sequences. ∆

(3)

A•A

∆31

A•A•A

A ∆

A•A

∆32

Combined with Proposition 2.10, this generalizes to the following. ˜ be a symmetric sequence with cocompoProposition 3.10. Let (A, ∆) sition. Then the transformation ∆ni : A•(n−1) → A•n equalizes the two transformations •n ∆n+1 , ∆n+1 ⇒ A•(n+1) . i i+1 : A More generally, ∆n+1 ∆ni = ∆n+1 ∆nj−1 for j > i. j i ˜ be a symmetric sequence with cocomposiCorollary 3.11. Let (A, ∆) tion. There are canonical, unique maps ∆[n] : A → A•n . (Given by taking any chain of compositions ∆nin · · · ∆1i1 .)

3.4

Counit and Codegeneracies

Write 1 for the functor 1 : Σ∗ → C given by # 1⊗ if |T | = 1, 1(T ) = ⋆C otherwise. We will call 1 the “counit” symmetric sequence. [The dual definition of the “unit” symmetric sequence would use ∅C .] Definition 3.12. A counital symmetric sequence is (A, ϵ˜) where A is a symmetric sequence and ϵ˜ is a natural transformation to the counit ϵ˜ : A → 1. Note that being counital is equivalent to the existence of a map A(1) → 1⊗. We will not require the map A(1) → 1⊗ to be equipped with a section. In the next subsection, we will use the following basic equality whose proof can be read oﬀ of Figure 5.

Cooperads as Symmetric Sequences

Lemma 3.13. The following functors Σ∗ → C are equal. ! " ! " 1 ! A s10 = A = A ! 1 s11 .

More generally, the following functors Σ≀n ∗ → C are equal. #! " ! "$ A!i ! 1 ! A!(n−i) sni = A!n

In the footsteps of Lemma 3.13 we define the following generalization. Definition 3.14. Given a counital symmetric sequence (A, ϵ˜) define associated natural transformations ϵ˜ni : A!(n+1) sni → A!n , for 0 ≤ i ≤ n, to be the following compositions. A!(n+1) sni #!

A!n

" ! "$ A!i ! A ! A!(n−i) sni

(Id!˜ ϵ!Id) sn i

Define ϵ˜00 = ϵ˜ : A → 1.

#! " ! "$ A!i ! 1 ! A!(n−i) sni

These natural transformations induce codegeneracies in the following manner. Since γ n+1 sni = γ n , the universal transformation A•(n+1) γ n+1 → A!(n+1) ! " induces a transformation A•(n+1) → Rγ n A!(n+1) sni . A•(n+1) → A•n to be the following composition. (4)

A•(n+1)

! " Rγ n A!(n+1) sni

Rγ n (˜ ϵn i)

Define ϵni :

A•n ! " Rγ n A!n

Similar to Proposition 3.10, the corresponding properties of sni imply the following. Proposition 3.15. Let (A, ϵ˜) be a counital symmetric sequence. Then the transformation ϵin−1 : A•n → A•(n−1) coequalizes the two transforn−1 n mations ϵni , ϵni+1 : A•(n+1) ⇒ A•n . More generally ϵin−1 ϵnj = ϵj−1 ϵi for j > i.

B. Walter

3.5

Cooperads and Cosimplicial Structure

Definition 3.16. A cocomposition operation on a counital symmetric sequence respects the counit if the following diagram of natural transformations Σ∗ → C commutes. =

(5)

Aγ 2 s10

˜ 1 ∆s 0

(A ! A)s10

A =

(˜ ϵ!Id)s10

(1 ! A)s10

Aγ 2 s11

˜ 1 ∆s 1

(A !

A)s11

(Id!˜ ϵ)s11

(A !

1)s11 =

A counital cooperad is a counital symmetric sequence with cocomposition which respects the counit. By applying Proposition 2.10 and using the simplicial structure of wreath product categories, the requirement in Definition 3.16 implies a more general statement. ˜ ϵ˜) is a cooperad, then the following comProposition 3.17. If (O, ∆, position is equal to the identity IdO!n , for j = (i − 1), i. ˜ n+1 sn ∆ j i

ϵ˜n j

O!n = O!n ∂in+1 snj −−−−−−→ O!(n+1) snj −−→ O!n Furthermore, the following compositions are equal if j < i − 1. O!n (∂in+1 snj ) n ) O!n (sjn−1 ∂i+1

˜ n+1 sn ∆ j i

n ϵ˜n−1 ∂i+1 j

O!(n+1) snj n O!(n−1) ∂i+1

ϵ˜n j

˜n ∆ i+1

O!n O!n

as well as the similar statement for j > i. We have now almost completed the proof of the following. ˜ ϵ˜) is a cooperad, then the collection {O•n }n Theorem 3.18. If (O, ∆, along with coface maps ∆ni and codegeneracy maps ϵni defines a coaugmented cosimplicial symmetric sequence with two extra codegeneracies. (6)

O•O

O•3

O•4

···

Cooperads as Symmetric Sequences

Proof. In Propositions 3.10 and 3.15, we have already shown the cosimn−1 n ϵi . ∆nj−1 and ϵin−1 ϵnj = ϵj−1 plicial identities ∆n+1 ∆ni = ∆n+1 j i n+1 n It remains only to consider the compositions ∆i ϵj . These come from the right Kan extension over γ n of the statements of Proposition 3.17. Note that the right Kan extension " ! ˜ n+1 sn ∆ j i Rγ n O!n ∂in+1 snj −−−−−−→ O!(n+1) snj

is equal to the composition

# # $ ∆n+1 $ # $ Rγ n+1 O!n ∂in+1 −−−i−−→ Rγ n+1 O!(n+1) −→ Rγ n O!(n+1) snj . Corollary 3.19. There are unique transformations ∆[n] : O → O•n . These are equal to any combination of parenthesization maps and cocomposition maps from their source to their target.

Cooperads via • versus !

We will now connect the cooperad structures defined in the previous sections with the classical methods which would attempt to use only the induced product • on symmetric sequences. When using •, the lack of associativity introduces extra “parenthesization” maps, which must be dealt with carefully.

4.1

Parenthesization Maps.

˜ ϵ˜) From now on, let A, B, C be generic symmetric sequences and (O, ∆, be a generic counital cooperad. Proposition 4.1. There are canonical “parenthesization” natural transformations: (A • B) • C A•B•C A • (B • C) More generally there are parenthesization maps to A1 •· · · •An from any parenthesization of this expression.

B. Walter

Proof. We show the existence of the map (A • B) • C → A • B • C. The other maps are similar. The universal natural transformation (A•B) γ 2 −→ (A!B) induces a natural transformation of functors (Σ∗ ≀ Σ∗ ) ≀ Σ∗ −→ C: ! " (A • B) ! C ∂13 −→ (A ! B) ! C = A ! B ! C.

The desired map is induced by taking the right Kan extension Rγ 3 of the diagram above. (A • B) • C Rγ 3

A•B•C

#! " $ (A • B) ! C ∂13

Rγ 3 (A ! B ! C)

Remark 4.2 (On the associativity of •). Without making further assumptions, it is not true that (A • B) • C ∼ = A•B•C ∼ = A •! (B • C). This would follow from the existence of natural equivalences Rγ 2 (A ! " (A ! B ! C) as well as the corresponding equivalence B) ! C ∼ R 3 = ∂1 using ∂23 . However, this will generally only occur in the unlikely event of the symmetric monoidal product commuting with categorical products. The situation contrasts starkly with that of the operad composition product, defined dual to • using left rather than right Kan extensions. If C is a closed monoidal category, then ⊗ is a left adjoint, so it will in particular commute with categorical coproducts and left Kan extensions. In this case the parenthesization maps for the operad composition product are isomorphisms and the operad composition product is associative. In practice, authors have generally dealt with this in the past by either not using cooperads at all, or by (implicitly or restricting %explicitly) & their categories so that (· · · )Σn = (· · · )Σn and = . In this (very special case) A • B = A ◦ B and there is no problem. Alternately, heavy restrictions can be placed on C and/or %on the category of symmetric sequences to force ⊗ to commute with . For example, in the category of ˆ finitely generated, injective bimodules over a commutative ring, ⊗ = ⊗ which is a right adjoint. Proposition 4.3. Parenthesization maps are associative. For example the following diagrams commute.

(7)

! " (A • B) • C • D

(A • B • C) • D A•B•C •D (A • B) • C • D

Cooperads as Symmetric Sequences

(A • B) • C • D (8)

(A • B) • (C • D)

A•B•C •D A • B • (C • D)

Proof of 4.3. It is enough to consider Diagrams (7) and (8). Commutativity is shown by writing the diagrams as right Kan extensions. The diagrams above are Rγ 4 of the following diagrams of functors Σ≀4 ∗ → C. (7’) ! " (A • B • C) ! D (γ 3 ≀ Id) #! $ " (A • B) • C ! D (γ 3 ≀ Id) A!B!C !D ! " (A • B) ! C ! D ∂14 (8’)

" (A • B) ! C ! D ∂14

! " (A • B) ! (C • D) (γ 2 ≀ Id ≀ γ 2 ) A!B!C !D ! " 4 A ! B ! (C • D) ∂3

Diagram (7’) is just − ! D applied to the following universal diagram (in which the upper-left map is Rγ 3 of the lower-right). (A • B • C) γ 3 (7”)

" (A • B) • C γ 3

A!B!C ! " (A • B) ! C ∂13

Diagram (8’) commutes because the upper and lower composition are both equal to ! " ι !ι (A • B) ! (C • D) (γ 2 ≀ Id ≀ γ 2 ) −−−1−−2−→ (A ! B) ! (C ! D)

Where ι1 : (A • B) γ 2 → A ! B and ι2 : (C • D) γ 2 → C ! D are the universal natural transformations from their respective Kan extensions.

4.2

Cooperad Structures

We relate parenthesization maps with cooperad structure. By the functoriality of !, there are natural transformations Id ! ∆ : A ! O → A ! (O • O) and ∆ ! Id : O ! A → (O • O) ! A,

B. Walter

where A is any symmetric sequence. Define the maps Id • ∆ and ∆ • Id to be the natural transformations induced on right Kan extensions via functoriality of Kan extension. For example ! " ∆ • Id = Rγ 2 ∆ ! Id : O • A −→ (O • O) • A.

By alternately letting A be a parenthesization of O•k and using functoriality of • this defines maps originating in any parenthesization of O•n . For example !

" ! " ! " (Id • Id) • ∆ • Id : (O • O) • O • O −→ (O • O) • (O • O) • O.

Theorem 4.4. The following diagrams commute (unlabeled maps are parenthesization): (O • O) • O

∆•Id

O•O

Id•∆

O•O•O

∆31

O•O

O • (O • O) ∆32

O•O•O

More generally, parenthesization maps convert Id•∆•Id (and its parenthesizations) to ∆42 , etc. Proof. We show the first diagram commutes. The second diagram and more general statement are proven the same. Consider the diagram below, where maps marked ι are all universal ι transformations of right Kan extensions (RF X) F −→ X. (9) ! " ! " ι ∂13 (O • O) • O γ 3 (O • O) ! O ∂13 !

(∆•Id) γ 3

O • O γ3

ι ∂13

∆31 γ 3

1 ⃝

(∆!Id) ∂13

(O ! O) ∂13 3 ⃝

(O • O • O) γ 3

2 ⃝

ι!Id

˜ ∆!Id ι

O!O!O

1 and ⃝ 3 commute by functoriality of right Kan exParallelograms ⃝ 1 is Rγ 2 of the right side, and tension. The left side of parallelogram ⃝ 3 is Rγ 3 of the right side. Triangle ⃝ 2 the left side of parallelogram ⃝ ˜ commutes by functoriality of ! (recall that ι∆ = ∆). Applying Rγ 3 along the outside of Diagram (9) yields the following

Cooperads as Symmetric Sequences

(where the map labeled ∗ is the parenthesization map): =

(O • O) • O

∆•Id

(10)

∗

O•O ∆31

O•O•O

Example 4.5. The following diagram is commutative (the unlabeled maps are parenthesizations): (∆•Id)•Id

(O • O) • O

∆•Id•Id

O•O•O

! " (O • O) • O • O (O • O) • O • O

O•O•O•O

∆41

Theorem 4.4 has the following corollary: Corollary 4.6. Commutativity of the following diagrams are equivalent. ˜ ∆

˜ ∆≀Id

Aγ 3

(11)

A!A!A ˜ ∆

∆

(12)

(A ! A)(γ 2 ≀ Id) (A ! A)(Id ≀ γ 2 )

A•A

∆•Id

˜ Id≀∆

(A • A) • A A•A•A

A ∆

A•A

Id•∆

A • (A • A)

Thus a cooperad could equivalently be defined as (O, ∆, ϵ) where O is a symmetric sequence, ∆ : O → O • O so that the analog of Diagram (12) commutes, and ∆ is compatible with the counit ϵ : O(1) → 1⊗ .

Comodules and Coalgebras

˜ O , ϵ˜) be a counital cooperad and M be Throughout this section, let (O, ∆ a symmetric sequence. The definition of cooperad comodules mirrors that of coalgebra comodules (dual to algebra modules). The benefit of viewing cooperads as symmetric sequences is that coalgebras over a cooperad can be viewed, essentially, as a special type of comodule.

B. Walter

5.1

Comodules

˜ M ) where M is a symmetric Definition 5.1. A left O-comodule is (M, ∆ ˜ M : M γ2 → O ! M is compatible with ∂ 3 and ∂ 3 and sequence and ∆ 1 2 s10 . That is, the following diagrams (analogous to Diagrams (1) and (5)) should commute. ˜M ∆

M γ3

(13)

˜ ∆!Id

O!O!M ˜M ∆

(14)

(O ! M ) (γ 2 ≀ Id)

M γ 2 s10

(O ! M ) (Id ≀ γ 2 ) ˜ M s1 ∆ 0

(O ! M ) s10

˜M Id!∆

(˜ ϵ!Id) s10

(1 ! M ) s10

As with cooperads, we write ∆M for the induced universal transformation to the right Kan extension ∆M : M" → O • M . There are ! !(n−1) n+1 ˜ induced transformations ∆i : O ! M ∂in+1 → O!n ! M and n+1 : O•(n−1) • M → O•n • M . ∆i Theorem 5.2. Analogous to Theorem 3.18 there is a canonical coaugmented cosimplicial complex as below. M

O•M

O•2 • M

O•3 • M

···

[n]

Corollary 5.3. There are unique transformations ∆M : M → O•(n−1) • M . These are equal to any combination of parenthesization maps and cocomposition maps from their source to their target. Remark 5.4. Right O-comodules could be defined similarly. However recent experience suggests that right comodules are most interesting in the non-counital case; in which situation we should use partial cocomposition products rather than cocomposition products. This moves beyond the scope of the current work.

5.2

Coalgebras

Let a be an object of C and A be a symmetric sequence. Note that a can be viewed as a functor a : Σ0 → C. Recall the descriptions of the

Cooperads as Symmetric Sequences

category ˆ ∅n in Remarks 2.16 and 2.7. We may view ˆ∅n either as the ≀(n−1) ≀(n+1) category of level n trees with no leaves; or as Σ∗ ⊂ Σ∗ , the full subcategory consisting of chains of set maps of the following form. fn−2

⋆ ←− S1 ←− · · · ←−−− Sn−1 ←− ∅ ≀0

Note that the category Σ∗ consists of only the trivial chain (⋆ ← ∅). This is equivalent to Σ0 . ≀(n+1) The face and degeneracy maps of Σ∗ induce the following face ≀(n−1) and degeneracy maps on Σ∗ . (We introduce an index shift below ≀(n−1) n n so that ∂¯i and s¯j map from ˆ ∅n = Σ∗ .) ! ≀(n−1) ≀(n−2) ∂¯in : Σ∗ → Σ∗ , for 1 ≤ i ≤ (n − 1), and n > 1 ≀(n−1)

s¯ni : Σ∗

≀n

→ Σ∗ ,

for 0 ≤ i ≤ n and n ≥ 1

The degeneracy map s¯nn doubles ∅, recognizing that a tree without leaves ≀1 of level n is also of level (n + 1). Note that ∂¯12 : Σ∗ → Σ0 coequalizes all ≀(n−1) ≀1 chains of face maps from Σ∗ to Σ∗ . We write γ¯ n for the composition γ¯ n = (∂¯i22 · · · ∂¯inn ). ≀n

≀(n+2)

, Definition 3.2 of symmetUnder the identification Σ∗ ⊂ Σ∗ ric sequence composition restricts to a functor (A1 ! · · · An−1 ! a) : ≀(n−1) Σ∗ → C. For example, A ! a is given by the following. $ & % " f0 f1 # a(∅) ⋆ ←− S ←− ∅ (−→ A(S) ⊗ s∈S ⊗|S|

= A(S) ⊗ a

The right Kan extension of Definition 3.2 restricts to a right Kan ex≀(n−1) tension over γ¯n : Σ∗ → Σ0 , yielding the following functor. A1 • · · · • An−1 • a = Rγ¯ n (A1 ! · · · ! An−1 ! a) : Σ0 −→ C ) Σk '( ⊗k For example, (A • a) = . A(k) ⊗ a k≥0

Remark 5.5 (Completed tensor product). In categories with a comˆ the completed coendomorphisms coend(A) ! pleted tensor product ⊗, = * ˆ ⊗n Hom(A, A ) form a cooperad. Cocomposition is dual to the comn position operation on the operad end(A∗ ). Dualizing the classical algebra definition, coalgebras should be equivalent to objects equipped with

B. Walter

! cooperad maps coend(A) → O. The completed tensor should be a right ˆ ˆ A⊗n adjoint to Hom so that this is equivalent to maps A → O(n) ⊗ . In ˆ this case, the definition of coalgebras above should use ⊗ rather than ⊗ (though ⊗ should still be used for the cooperad). A more detailed survey of this issue is beyond the scope of the current work, which is intended to focus on cooperads. ˜ ϵ˜) is (c, ∆ ˜ c ) where Definition 5.6. A coalgebra over the cooperad (O, ∆, 2 ˜ c is an object of C and ∆c : c γ¯ → O ! c is compatible with face maps ∂¯12 = (γ 2 ≀ Id), ∂¯22 = (Id ≀ γ¯ 2 ) and degeneracy s¯10 . That is, the following diagrams (analogous to Diagrams (13) and (14)) should commute. ˜c ∆

(O ! c)(γ 2 ≀ Id)

c γ¯ 3

(15)

O!O!c ˜c ∆

(16)

˜ ∆!Id

c γ¯2 s¯10

(O ! c)(Id ≀ γ¯2 )

˜ c s¯1 ∆ 0

(O ! c) s¯10

˜c Id!∆

(˜ ϵ!Id) s¯10

(1 ! c) s¯10

Statements and proofs about left comodules translate into statements and proofs about coalgebras by converting ∂in , sni into ∂¯in , s¯ni . Essentially, coalgebras are left comodules which are concentrated in 0arity. Write ∆c for the induced map (in "C) ∆c : c → O • c. As with ! !(n−1) n+1 ˜ comodules we have ∆i : O ! c ∂¯in+1 → O!n ! c inducing n+1 •(n−1) •n ∆i :O • c → O • c. Theorem 5.7. The comultiplication ∆c defines a canonical coaugmented cosimplicial complex (in C) c

O•c

O•2 • c

O•3 • c

···

Corollary 5.8. There are unique C-maps ∆[n] : c −→ O•(n−1) •c. These are equal to any combination of parenthesization maps and cocomposition maps from their source to their target.

Cooperads as Symmetric Sequences

Examples

We end with a two simple examples of cooperads which are not duals of standard operads. Both of these are constructed via quotient/contraction operations. The (directed) graph cooperad is used in [9] and the contractible ∆ complex operad is a generalization.

6.1

The Graph Cooperad

Given a finite set S, a contractible S-graph is a connected, acyclic graph whose vertex set is S. The unoriented graph cooperad has gr(S) equal to the free Z module generated by all contractible S-graphs. The co˜ : gr γ 2 → gr ! gr is defined as composition natural transformation ∆ follows. Given two graphs G and K, a quotient map of graphs q : G " K is a surjective ! map from " vertices of G onto vertices of K such that q(v1 , v2 ) = q(v1 ), q(v2 ) defines a map sending edges of G to edges and vertices (if q(v1 ) = q(v2 )) of K, surjecting onto the edges. Note that if q : G " K is a quotient map and v is a vertex of K, then q −1 (v) is a subgraph of G. A graph contraction is a quotient map where each q −1 (v) is a connected subgraph. Note that there is a bijection between the edges of G and the edges of K union those of the q −1 (v). Suppose G is an S-graph and f : S " T is a surjection of sets. Given t ∈ T , let f −1 (t) be the maximal subgraph of G supported by the vertices of f −1 (t). We say that f induces a graph contraction on G if f −1 (t) is contractible for each t. In this case, we define the induced contracted graph (G/f ) to have vertices T with an edge from vertex t1 to t2 if there is an edge in G from the subgraph f −1 (t1 ) to the subgraph f −1 (t2 ). ! f " ˜ takes the element T ← Cocomposition ∆ − S of Σ∗ ≀ Σ∗ to the map #$ % gr(S) −→ gr(T ) ⊗ gr(f −1 (t)) t∈T

" −1 (t) if f defines a graph which takes a S-graph G to (G/f ) ⊗ t∈T f contraction on G, and sends G to 0 otherwise. Since the quotient operation described previously is clearly associative, this defines a symmetric sequence with cocomposition. The counit map sends S-graphs with only one vertex to 1 ∈ Z and kills all others. The (directed) graph cooperad is similar to the unoriented graph cooperad. In the category of directed, contractible S-graphs define

B. Walter

− ⇀ gr(S) = gr(S)/ ∼, where ∼ identifies reversing the orientation of an edge with multiplication of a graph by −1. Cocomposition on gr gives − ⇀ a well-defined map on gr since reversing an arrow in G will reverse exactly one arrow either in the quotient graph G/f or in one of the f −1 (t). Free nilpotent coalgebras over the graph cooperad can be written as free Z modules generated by all (finite) graphs with vertices labeled by the primitive elements. The cooperad structure operates by ripping ! f " ˜ takes T ← out subgraphs. Explicitly ∆ − ∅ to the map which takes a " !# −1 (t) taken over all set f graph G to the sum of terms (G/f ) ⊗ t∈T maps f : Vert(G) → T . [With the convention that G/f = 0 if f does not define a graph contraction on G.] Note that f ˜ ∆(T ← − ∅)

will kill a graph G for all sets T with |T | > |Vert(G)| (because graph contraction maps cannot increase the number of vertices). Thus this coalgebra is nilpotent. We will leave the full proof that this is a free nilpotent coalgebra for the sequel. The graph cooperad generalizes to the following.

6.2

The CDC Cooperad

By a ∆-complex, we mean what Hatcher [8, Appendix] calls a “singular ∆-complex” or “s∆-complex”. Essentially this is a CW complex whose cells are all (oriented) simplices and whose attaching maps factor through face maps of the simplex. Given a set S, an S ∆-complex is a ∆-complex whose 0-cells are labeled by elements of S. The CDC cooperad has cdc(S) equal to the free Z module generated by contractible S ∆-complexes. Cocomposition is defined similar to that for gr. If T is a subset of the 0-cells of a ∆-complex X, write T for the maximal CW subcomplex of X supported by T . Quotient maps for ∆-complexes are CW quotient maps. We say a quotient map X ! Y is a contraction if the inverse image of each 0-cell of Y is a contractible subcomplex of X. If X is a S ∆-complex then a set surjection f : S ! T induces a CW contraction on X if f −1 (t) is contractible for each t ∈ T . In this case, we define (X/f ) to be the quotient of X by the sub CWf

complexes f −1 (t). The cocomposition map of cdc takes − S)" to !# (T ← −1 (t) if f the map which sends the S ∆-complex X to (X/f ) ⊗ t∈T f induces a CW contraction on X and 0 otherwise.

Cooperads as Symmetric Sequences

Acknowledgments I would like to thank Dev Sinha, whose questions led to the inception of this work; as well as Michael Ching who resolved many of my early confusions. Also Clemens Berger, Bruno Vallette, and Jim McClure listened to early versions of these ideas and provided invaluable feedback. Most of all, I must thank Kallel Sadok and the Mediterranean Institute for Mathematical Sciences (MIMS) for an invitation to speak at the conference on “Operads and Configuration Spaces” in June 2012, which led me to finally revising and clarifying these ideas which have been on paper and bouncing around in my head for almost six years. This work is based on the notes from my series of talks at MIMS. Benjamin Walter Mathematics Research and Teaching Group, Middle East Technical University, Northern Cyprus Campus, Kalkanli, G¨ uzelyurt, KKTC via Mersin 10, Turkey benjamin@metu.edu.tr

References [1] Mahowald Arone; Greg; Mark, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), 743–788. [2] Clemens B., Iterated Wreath Product of the Simplex Category and Iterated Loop Spaces, arXiv (2005). [3] Block R., Recognizable formal series on trees and cofree coalgebraic systems, Journ. Alg. 215 (1999), 543–573. [4] Ching M.,Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005), 833–933. [5] Ching M., A note on the composition product of symmetric sequences, arXiv:math/0510490v2 (2012). [6] Fox T., The construction of cofree coalgebras, JPAA 84 (1993), 191–198. [7] Hazewinkel M., Cofree coalgebras and multivariable recursiveness, JPAA 183 (2003), 61–103.

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[8] Hatcher A., Algebraic Topology, Cambridge Univ. Press, (2002). [9] Sinha D.; Walter B., Lie coalgebras and rational homotopy theory, I: Graph coalgebras, Homology, Homotopy and Applications 13: 2, (2011), 1–30. [10] Sinha D.; Walter B., Lie coalgebras and rational homotopy theory, II: Hopf invariants, Trans. Amer. Math. Soc 365: 2 (Feb. 2013), 861–883. [11] Smith J., Cofree coalgebras over operads, Top. and Appl., 133 (2003), 105–138. [12] Walter B.,Cofree coalgebras over cooperads, in preparation [13] Walter B., Lie algebra configuration pairing, arXiv:1010.4732, (2010).

Morfismos, Vol. 17, No. 2, 2013, pp. 101–125

Moduli spaces and modular operads Jeﬀrey Giansiracusa

Abstract We describe a generalised ribbon graph decomposition for a broad class of moduli spaces of geometric structures on surfaces (with nonempty boundary), including moduli of spin surfaces, r-spin surfaces, surfaces with a principle G-bundle, surfaces with maps to a background space, surfaces with Higgs bundle, etc.

2010 Mathematics Subject Classification: 57M50, 57M15, 18D50, 18D05, 58D27. Keywords and phrases: Ribbon graphs, moduli spaces, mapping class group, arc complex, 2-categories, cyclic operads.

Introduction

This paper is an expansion of some ideas that I first talked about in 2012 in the MIMS conference on Operads and Configuration Spaces. Here I shall give a more detailed account, though still not a complete one, of a certain theorem about modular envelopes. The full details will appear in a future paper; in this note I will try to be expository and focus on illuminating the central ideas without being overly concerned by technical details that might otherwise obscure some of the conceptual clarity of the arguments. Fix a class ψ of geometric structures on surfaces. For example, one could take orientations, principal G-bundles, or spin structures, etc. Associated to any surface Σ is the space ψ(Σ) of all such structures on that surface. Taking the homotopy quotient by the diﬀeomorphism group yields a homotopy theoretic moduli space of surfaces with ψ-structure. If we consider surfaces with some marked intervals along the boundary, and ψ-structures that have a fixed value on each marked interval, then we can glue the intervals together and the result is a modular operad. 101

102

Jeﬀrey Giansiracusa

denoted Mψ . (If, instead of a single fixed value on the intervals, we allow one of several fixed valued then the result is instead a coloured modular operad). These moduli spaces are the objects we wish to study. The idea of this work is to decompose them, in a sense, into moduli spaces of discs with ψ-structure. The modular operad Mψ contains a sub-cyclic operad Dψ of moduli spaces of discs with ψ-structure. Our main result is that Mψ is freely generated (in a homotopical sense) as a modular operad over this sub-cyclic operad. I.e., the derived modular envelope of Dψ is weakly equivalent to Mψ . This result was inspired by the work of Costello. As part of his groundbreaking work in the homotopy theory of open-closed topological field theories [9], he gave a new perspective on the very important idea of describing the moduli space of Riemann surfaces with ribbon graphs in [8, 10]. He proved that the derived (i.e., homotopy invariant) modular envelope of the associative operad gives a model for the modular operad of moduli spaces of Riemann surfaces with open-string type gluing for the compositions. A point in this modular envelope can be described as a graph equipped with lengths on all of its edges and a cyclic order of the edges incident at each vertex — i.e., a metric ribbon graph. Thus the moduli space of ribbon graphs is equivalent to the moduli space of Riemann surfaces.

Costello’s proof used geometry and analysis on a certain partial compactification of the moduli space of Riemann surfaces. Thus it appears his argument is not suited to more homotopy theoretic contexts such as the one considered in this paper. In [13], I gave a diﬀerent proof of Costello’s modular envelope theorem. This proof instead rested on the well-known contractibility of the arc complex of a surface. This new argument lead to an adaptation to dimension 3: the derived modular envelope of the framed little 2-discs is equivalent to the modular operad of moduli spaces of 3-dimensional handlebodies. Here we instead focus of refining and generalising the argument of [13] in dimension 2. When the structures being considered are principal G-bundles then we expect this result will lead to a G-equivariant version of Costello’s open-closed TFT theorem.

Operads

A cyclic operad in C is a functor P from the category of finite sets and bijections to C together with composition maps

Moduli spaces and modular operads

103

i ◦j

P(I) ⊗ P (J) → P(I # J ! {i, j}), for i ∈ I and j ∈ J, satisfying an associativity condition and natural in (I, i) and (J, j). One can think of P as a collection of abstract “electrical circuit components,” where P(I) as a set/space of components with terminals given by the set I. The composition maps correspond to wiring terminals together to produce new components; terminals can only be glued in pairs (no trivalent connections) and in a cyclic operad two components can only be glued together in at most one place. Allowing multiple gluings leads to the following definition. A modular operad in C is a cyclic operad Q together with natural ◦i,j self-composition maps Q(I) → Q(I ! {i, j}) that commute with the cyclic operad composition maps and with each other. Example 2.1. 1. The commutative modular operad is the constant functor sending each finite set to a point. 2. The associative cyclic operad Assoc sends I to the set of cyclic orders on I. We will need a slight generalisation in which there are diﬀerent types of terminals and two terminals can only be connected if they are of the same type. The types are called colours. Fix a set Λ, which we will call the set of colours. A Λ-coloured set I consists of a finite set with a map to Λ. A morphism I → I ′ of coloured sets is a bijection that respects the colours. A coloured cyclic operad P is a functor from the category of coloured sets to C together with a collection of composition maps i ◦j as before, but now only defined when i and j have the same colour. A coloured modular operad is defined analogously, where the self-composition maps ◦ij also only defined when i and j have the same colour.

2.1

Homotopy theory of cyclic and modular operads

Berger and Batanin [7] have recently constructed fully satisfactory Quillen model category structures on cyclic and modular operads. When talking about derived constructions such as the derived modular envelope, one could work with the model category structures. However, we take a more pragmatic approach, since the modular envelope is the only functor we ever have to derive, and our construction of the derived functor will be manifestly homotopy invariant due to the homotopy invariance of

104

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homotopy colimits. We need only the following definition. A morphism P → P ′ of cyclic or modular operads in spaces is a weak equivalence if each map of spaces P(I) → P ′ (I) is a weak equivalence.

3 3.1

Category theory and homotopy theory The nerve of a category

Let C be a category, which we assume is small, meaning that the objects form a set rather than a class. (E.g., The category of all sets is not small, but the category of all subsets of a fixed big set is small.) We can associate a simplicial set (and hence a space) with C via the nerve construction. The nerve of C , denoted N (C ) is the simplicial set whose 0-simplices are the objects of C , 1-simplices are the morphisms of C , 2-simplices are the 2-simplex shaped diagrams in C X0 g◦f

" # X2 ,

and so on. In general, the n-simplices N (C )n are the set of composable n-tuples of morphisms, f1

X0 → X 1 → · · · → X n . We will write BC for the geometric realisation of the nerve. It is easy to see that a functor F : C → D induces a map of nerves N• C → N• D. A natural transformation F → F ′ induces a homotopy between the corresponding maps. From this it follows that an adjoint pair (F, G) induces a homotopy equivalence of nerves. If a category C has an initial object u then there is a natural transformation form the constant functor with value u to the identity, and so the nerve of C is contractible. Likewise, existence of a final object implies contractibility of the nerve.

3.2

The fundamental group of the nerve of a category

While computing the higher homotopy groups of a space is usually very diﬃcult, there is a convenient recipe for computing the fundamental group of the nerve of a category.

Moduli spaces and modular operads

105

Given C , let C [C −1 ] denote the category formed by adjoining inverses to all of the arrows in C (see [6, §1.1]) This localisation is clearly a groupoid (all morphisms are invertible). If two objects of a groupoid lie in the same connected component then their automorphism groups are isomorphic (by an isomorphism that is unique up to conjugation). Given an object x of C , the fundamental group of N C based at x is canonically isomorphic to the automorphism group of x in the groupoid C [C −1 ].

3.3

Strict 2-categories

A strict 2-category is a category enriched in Cat. I.e., it consists of a class of objects ObjC , a category HomC (a, b) for each pair of objects, and composition functors HomC (a, b) × HomC (b, c) → HomC (a, c) that are strictly associative and for which a unit exists in HomC (a, a). The objects of the hom categories are called 1-morphisms and the morphisms of the hom categories are called 2-morphisms. Example 3.1. Let T op 2 denote the strict 2-category whose objects are spaces, and for which Hom(X, Y ) is the groupoid of maps and homotopy classes of homotopies. For example, the groupoid of morphisms from a point to a circle is equivalent to the group Z (i.e., one object and automorphism group Z). Example 3.2. Let Cat 2 denote the strict 2-category whose objects are small categories and whose hom categories are the categories of functors and natural transformations. Example 3.3. Since a set can be considered as a category with no non-identity morphism, an ordinary category can be considered as a 2-category in which there are no non-identity 2-morphisms. Remark 3.4. In this paper, all strict 2-categories that arise will have the property that all 2-morphisms are in fact isomorphisms. Such a 2-category is sometimes called a (2,1)-category. Strict 2-categories are a restricted class of 2-categories. More generally, one often wants to work with weak or lax 2-categories, where the associativity and unit conditions only hold up to natural transformations (which must then satisfy some conditions). We will have no need of these more sophisticated notions in this paper.

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Jeﬀrey Giansiracusa

A strict 2-functor between strict 2-categories F : C → D is a map F : ObjC → D together with a functor Hom(a, b) → Hom(F (a), F (b)) for each pair of objects such that these functors are strictly compatible with the composition functors. We will usually abbreviate and call this a functor. We will leave it as an exercise to spell out precisely what a natural transformation between strict 2-functors is. A strict 2-category C has a nerve N (C ) that is a bisimplicial set. It is constructed as follows. First one replaces all the hom categories with their nerves to obtain a simplicial category. Then the nerve of this simplicial category yields a bisimplicial set that is the nerve of C . As in the case of 1-categories, a strict 2-functor induces a map of nerves, and a natural transformation induces a homotopy. In particular, observe that if F : C → Cat 2 is a strict 2-functor then taking the realisation of the nerve pointwise yields a strict 2-functor C → T op 2 .

3.4

Over categories and Quillen’s Theorems A and B

Let F : A → B be a functor. Given an object b ∈ B, one can define a category of objects in A over b. This is denoted F ↓ b (or B ↓ b when F is the identity functor) and is called the over category of F based at b (some people instead call it the comma category). Its objects are pairs consisting of an object a ∈ A and a morphism g : F (a) → b in g′

B. A morphism from F (a) → b to F (a′ ) → b consists of a morphism h : a → a′ in A such that the diagram F (a) g

F (h)

" F (a′ ) g′

in B commutes. Observe that a morphism f : b → b′ in B induces a functor f∗ : (F ↓ b) → (F ↓ b′ ). There is also a canonical projection functor (F ↓ b) → A given by forgetting the morphism to b. Over categories can be thought of as a category-theoretic analogue of homotopy fibres. In fact, Quillen’s Theorems A and B are instances of this analogy. If the homotopy fibre of a map is contractible then the map is a weak equivalence. Theorem 3.1 (Quillen’s Theorem A). Let F : A → B be a functor and suppose that for each object b of B the nerve of the over category F ↓ b is contractible. Then F induces a weak equivalence of nerves.

Moduli spaces and modular operads

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This is a special case of a more general theorem that allows one to identify the homotopy fibre of a map of nerves induced by a functor. Theorem 3.2 (Quillen’s Theorem B). Let F : A → B be a functor and suppose for each morphism f : b → b′ in B the corresponding functor f∗ : (F ↓ b) → (F ↓ b′ ) induces a weak equivalence of nerves. Then (F ↓ b) → A → B is a homotopy fibre sequence. The construction of over categories and Quillen’s Theorems A and B have extensions to strict 2-categories. See [1] and [2] for details. Given a strict 2-functor F : A → B and an object x ∈ ObjB, there is an over 2-category (F ↓ x). It objects are pairs (a ∈ ObjA, f : F (a) → x). A f1

morphism from F (a1 ) → x to F (a2 ) → x consists of a morphism g : a1 → a2 in A and a 2-morphism in B from f1 to f2 ◦ F (g). A 1-morphism x → x′ in B induces a strict translation 2-functor (F ↓ x) → (F ↓ x′ ).

Theorem 3.3 (Theorem B for 2-categories,[2]). If all of the translation functors induce homotopy equivalences then N (F ↓ x) → N (A) → N (B) is a homotopy fibre sequence for any object x.

3.5

Left Kan extension

Let A , B, C be categories. Given a functor f : A → B, precomposition with f sends a functor B → C to a functor A → C . This defines a functor f ∗ : F un(B, C ) → F un(A , C ). It turns out that this f ∗ admits a left adjoint f! , which is called the left Kan extension.

3.6

Left Kan extensions and homotopy left Kan extensions

Let A , B and C be categories with C cocomplete. Consider functors A G

! C

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Recall that the left Kan extension of F along G is a functor G! F : B → C defined on objects by the colimit G! F (b) = colim F ◦ jb , (G↓b)

where (G ↓ b) is the comma category of objects in C over b and jb : (G ↓ b) → C forgets the morphism to b (to simplify the notation we will often omit writing jb ). Left Kan extensions possess a universal property: the functor G! F comes with a natural transformation F ⇒ G! F ◦ P that is initial among natural transformations from F to functors factoring through P . If C is a Quillen model category (such as topological spaces or chain complexes) then there is a homotopy invariant (or, derived) version known as the homotopy left Kan extension LG! F ; it is given by the formula LG! F (b) = hocolim F ◦ jb . (G↓b)

This construction is homotopy invariant in the following sense: a natural transformation F → F ′ that is a pointwise homotopy equivalence induces a natural transformation LG! F → LG! F ′ that is also a pointwise homotopy equivalence. In fact, this is the left derived functor of left Kan extension with respect to the projective model structure on the functor categories. There is a homotopy coherent version of the universal property for homotopy left Kan extensions. See [5, Proposition 6.1] for the details. Note that there is a “Fubini theorem” for both ordinary and homotopy colimits, colim F ∼ = colim G! F A

3.7

and

≃

hocolim F → hocolim LG! F. A

Homotopy colimits, the Grothendieck construction and Thomason’s Theorem

At several points we shall be taking homotopy colimits of diagrams in T op obtained from diagrams in Cat by applying the classifying space

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functor B (i.e. geometric realisation of the nerve) pointwise. Here we briefly recall a couple of useful tools for this situation. Given a functor F : C → Cat, the Grothendieck construction ! on F , denoted C F is the category in which objects are pairs (x ∈ C , y ∈ F (x)), and a morphism (x, y) → (x′ , y ′ ) consists of an arrow f ∈ homC (x, x′ ) and an arrow g ∈ homF (x′ ) (f∗ y, y ′ ). This construction ! satisfies an associativity condition: if F :!C → Cat and G : C F → ! Cat are functors then sending c ∈ ObjC to F (c) G defines a functor F G : C → Cat and there is a natural equivalence of categories $ " #" " G≃ G . ! C

Thomason’s Theorem [11, Theorem 1.2] asserts that there is a natural homotopy equivalence, #" $ ≃ F . hocolim BF −→ B C

As a special case, if C is actually a group G (a category with a single object ∗ and all ! arrows invertible), then BF (∗) is a space with a G action, and B( G F ) is homotopy equivalent to the homotopy quotient (BF (∗))hG . If C = ∆op semi then F ! is a semi-simplicial category, BF is a semisimplicial space, and B( ∆op F ) ≃ hocolim BF is equivalent to the semi geometric realisation of this semi-simplicial space. There is a 2-categorical version of the above. First of all, given a is a strict 2-category C and a strict 2-functor F : C → Cat 2 there ! Grothendieck construction that produces a strict 2-category C F over ! C . An object of C F is a pair (x ∈ ObjC , y ∈ ObjF (x)). A 1-morphism (x, y) → (x′ , y ′ ) is a pair (f1 , f2 ). where f1 : x → x′ is a 1-morphism in C and f2 : F (f1 )(y) → y ′ is a morphism in F (x′ ). A 2-morphism (f1 , f2 ) → (g1 , g2 ) consists of a 2-morphism α : f1 ⇒ g1 in C (which gives a natural transformation α∗ from F (f1 ) to F (g1 )) such that the diagram (in F (x′ )) F (f1 )(y) α∗

# g2

F (g1 )(y)

" y′

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commutes. This 2-categorical Grothendieck construction satisfies the obvious analogue of the associativity condition satisfied by the 1-categorical construction. Also, a 2-functor C → T op 2 has a homotopy colimit, and a 2-categorical version of Thomason’s theorem holds [2]: if F : C → Cat 2 is a 2-functor then ! hocolim BF ≃ B F. C

We again refer the reader to [2] and the references there for further details.

3.8

Graphs and Costello’s graph category

For us, a graph Γ will consist of a set V of vertices, a set H of half-edges, an incidence map in : H → V , and an involution ι : H → H, called the edge flip, that specifies how the half-edges are glued together. The free orbits of ι are the edges of the graph, denoted E(Γ), so each edge consists of a pair of half-edges. The fixed points are called the legs and are denoted L(Γ). The incidence map sends each half-edge to the vertex that it meets. A graph Γ has a topological realisation |Γ| as a 1-dimensional CW complex with a 0-cell for each vertex and a 1-cell for each edge and leg. A graph is a tree if its topological realisation is contractible, and a forest if it is a union of trees. A corolla is a graph that consists of a single vertex and a number of legs incident at it. If a graph is a disjoint union of corollas then the edge flip map is the identity and so giving a union of corollas is equivalent to giving a triple (V, H, in : V → H). Associated with a graph Γ are two disjoint unions of corollas. The first is given by forgetting the edge flip and is denoted ν(Γ) (cutting each edge into a pair of legs). The second is denoted π0 Γ; it has one vertex for each connected component of the graph and one leg for each leg of the original graph Γ. Costello [8] introduced a category Graphs in which the objects are disjoint unions of corollas and morphisms are given by graphs. In intuitive terms, we think of a morphism as assembling a bunch of corollas into a graph Γ followed by contracting all edges so that what remains is again a union of corollas (the result is π0 Γ). Composition of morphisms is defined by iterating this process. More precisely, the objects are triples, (V, H, in : V → H); a morphism from (V1 , H1 , in1 : V1 → H1 ) to (V2 , H2 , in2 : V2 → H2 ) is represented by a graph Γ together with an isomorphism from the source to ν(Γ) and an isomorphism from the target to π0 Γ. There is an obvious notion of equivalence on these data and the

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set of morphisms is defined as the set of equivalence classes. Alternatively, one can describe the set of morphisms as follows. A morphism consists of involution a on H1 so that (V1 , H1 , in1 : V1 → H1 , a) defines a graph Γ together with an isomorphism of π0 Γ with the union of corollas corresponding to the target. To define the composition of morphisms we use this second description. A composable pair of morphisms is given by a union of corollas, an involution a1 on the half edges, and then a second involution a2 on the set of fixed points of the first. The composition is given by the involution that is equal to a1 on the free orbits of a1 and is equal to a2 on the fixed points of a1 . Graphs also form a category in a diﬀerent way, where the objects are graphs and the morphisms are given by contracting a set of tree subgraphs to points. Disjoint union makes Graphs into a symmetric monoidal category. We will be interested in the symmetric monoidal subcategory Forests ⊂ Graphs which has only those objects containing no 0-valent components (i.e., no isolated vertices) and only those morphisms that are forests (i.e., disjoint unions of trees); the inclusion functor will be denoted ℓ. We will also be interested in the over category of this inclusion. Fix a union of corollas x and consider the over category ℓ ↓ x. Proposition 3.5. The category ℓ ↓ x is canonically equivalent to the category whose objects are graphs with legs identified with the legs of x and whose morphism are given by contracting a collection of disjoint trees down to points.

3.9

Coloured graphs

Fix a set Λ of colours. A Λ-coloured graph is a graph together with an element of Λ assigned to each edge and each leg. One can form a category of Λ-coloured graphs, generalised Costello’s category Graphs, in which the objects are disjoint unions of corollas and the morphisms are coloured graphs.

3.10

Cyclic and modular operads as functors

Costello introduced his categories of graphs in order to reformulate the definition of cyclic and modular operads in terms more amenable to doing homotopy theoretic constructions. Proposition 3.6. The category of cyclic operads in C is equivalent to the category of symmetric monoidal functors Forests → C , and

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the category of modular operads in C is equivalent to the category of symmetric monoidal functors Graphs → C . The idea is that a cyclic (or modular) operad P determines a functor by sending the n-corolla ∗n to the space P(n), and it sends a disjoint union of several corollas ∗n1 #· · ·#∗nk to the product P(n1 )⊗· · ·⊗P(nk ). Gluing legs together is sent to the map induced by the corresponding composition map. Coloured cyclic and modular operads can of course also be described as symmetric monoidal functors, using the categories of coloured graphs.

3.11

Some examples: the commutative and associative operads

We will mainly be concerned with the case when the ambient category in which our operads live is the category T op of topological spaces. The commutative operad Comm is the cyclic operad that is the constant functor Gr → T op sending each graph to a single point ∗. Clearly the commutative operad can also be considered as a modular operad. A ribbon structure on a graph is a choice of cyclic ordering of the half-edges incident at each vertex. A graph with ribbon structure is called a ribbon graph. The associative operad Assoc is the cyclic operad that sends each graph γ to the discrete space consisting of one point for each ribbon structure on γ. It is not hard to see that if γ → γ ′ is a contraction of a tree subgraph then there is a canonical bijection between ribbon structures on γ and on γ ′ . There is also a canonical bijection between ribbon structures on γ and on its atomisation, and this provides the natural isomorphism required in the definition of a cyclic operad.

3.12

Modular envelopes

Restriction from Graphs to Forests defines a forgetful functor from modular operads to cyclic operads. This functor admits a left adjoint, Mod, called the modular envelope. We think of the modular envelope of a cyclic operad as the modular operad it freely generates. The modular envelope can be constructed via left Kan extension along the inclusion Forests "→ Graphs. By replacing the Kan extension with the derived Kan extension, we have the derived modular envelope functor LMod.

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113

Moduli of geometric structures on surfaces Surfaces with collars

Let Σ be a surface with boundary and corners (by corners, we mean that it is locally modelled on the positive quadrant [0, ∞)2 ⊂ R2 ). The boundary of Σ is canonically partitioned into smooth strata, each of which is either a circle of an interval. A boundary interval is a boundary stratum that is an interval. Let J ⊂ Σ be a boundary interval. A collar of J is a smooth embedding φ of (−1, 0] × [0, 1] into Σ that sends boundary to boundary and is a diffeomorphism of {0} × [0, 1] onto J. A surface equipped with a finite set of disjoint boundary intervals equipped with disjoint collars is called a collared surface. If the collars are labelled by a set I then we call the surface I-collared. Suppose Σ1 and Σ2 are I-collared surfaces. A diffeomorphism of Icollared surfaces Σ1 → Σ2 is a diffeomorphism of the underlying surfaces that respects the labelling and parametrization of the collars. Suppose Σ is a surface with disjoint boundary intervals J1 and J2 equipped with disjoint collars φ2 and φ2 respectively. One can glue these two boundary intervals together and obtain a new smooth surface. This is done as follows: let Σ′ = Σ ! (J1 ∪ J2 )/ ∼, where we identify φ1 (x) with φ2 (x) for each x ∈ (−1, 0) × [0, 1].

4.2

Sheaves of geometric structures

Let Surf be the category enriched in T op of finite type surfaces (possibly with boundary and corners) and open embeddings. " is an enriched functor Definition 4.1. A smooth sheaf ψ on Surf " S urf → T op that sends pushout squares to homotopy pullback squares. Remark 4.2. Smooth sheaves of this type have been studied in [3], where they are called homotopy sheaves and their relation with GoodwillieWeiss embedding calculus of functors is explored. This notion also could go under the name of ∞-stacks. Here we will think of the space ψ(Σ) as the space of geometric structures of a given type on Σ. Below is a list of interesting examples of some of the kinds of structures that one can consider within this definition. Example 4.3.

1. Orientations: ψ(Σ) is the set of orientations on Σ.

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2. Almost complex structures: Since the space of almost complex structures (for a fixed orientation) is contractible, this is equivalent to simply taking orientations. 3. Principal G-bundles: The space associated with a surface Σ is the space of maps Σ → BG. 4. Maps to a background space X: The space ψ(Σ) is the space of maps Σ → X. 5. Spin, Spinc and r-spin can all be described in terms of the space of lifts of the classifying map Σ → BSO(2) of the tangent bundle. 6. Foliations: ψ(Σ) is the geometric realization of the simplicial space whose space of p-simplices is the space of codimension 2 foliations of Σ × ∆p that are transverse to the boundary of the simplex.1 One way to produce examples of smooth sheaves is to take sections of a bundle that is functorially associated with the tangent bundle. Let X be a space with an action of GL2 (R). Given a surface Σ, let P → Σ be the GL2 (R)-principal bundle associated with the tangent bundle and consider the bundle P ×GL2 (R) X → Σ. Proposition 4.4. Sending Σ to the space of sections of P ×GL2 (R) X !. (with the compact-open topology) defines a smooth sheaf ψX on Surf Similarly, if X is a smooth manifold on which GL2 (R) acts smoothly, then sending Σ to the space of smooth sections (with the smooth topology) defines a smooth sheaf. Those smooth sheaves arising in this way will be called tangential. Remark 4.5. A priori, the definition of a smooth sheaf appears more general than the definition of tangential smooth sheaf. Not every smooth sheaf is tangential, such as the example of foliations in the list above. However, every smooth sheaf admits a tangnetial approximation and sometimes the approximation is actually equivalent to the original sheaf. In more detail, as described in [4, p. 16–17], given an smooth sheaf ψ, there is associated a tangential sheaf τ ψ and a canonical comparison morphism ψ → τ ψ. Moreover, (a version of) Gromov’s h-principle gives conditions under which this comparison morphism is a weak equivalence of sheaves. 1

The author thanks the referee for suggesting the inclusion of this example.

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A smooth sheaf ψ will be called connected if ψ((−1, 0) × I) is connected. Assuming ψ is connected, we can choose a basepoint ∗ ∈ ψ((−1, 0) × I). Suppose J ⊂ ∂Σ is a boundary interval equipped with a collar φ. We say that a section s ∈ ψ(Σ) is trivial at J if φ∗ (s) restricts to the chosen basepoint ∗ on (−1, 0) × I. Remark 4.6. In proving the main theorem of this paper, the assumption that ψ is connected can be discarded if one is willing to work with coloured cyclic and modular operads instead of ordinary (single colour) cyclic and modular operads. If Σ is a surface equipped with a collection of disjoint collared boundary intervals {J1 , . . . , Jn }, we write ! ψ(Σ) ⊂ ψ(Σ)

for the subspace consisting of sections that are trivial at the boundary intervals Ji .

4.3

The monoid of geometric structures on a strip

Consider the unit square I × I equipped with a collar at each of the intervals {0} × I and {1} × I oriented in the same direction. We write Aψ for the space ψ ′ (I × I) of sections that are trivial at each side of the square because this space will play a particularly important role in the results ahead. Proposition 4.7. Gluing squares side to side endows the space Aψ with an A∞ composition making it into a group-like A∞ monoid; the homotopy inverse map is induced by rotating the square 180 degrees. Fixing a collared boundary interval J on a surface Σ, there is a right ! A∞ action of Aψ on ψ(Σ) by gluing the right side of a square to J, and a left A∞ action given gluing the left edge of a square to J. We will not spell out the proof of this here; it is straightforward but technical because of the necessity of using some machinery to handle A∞ monoids and their actions.

Proposition 4.8. Let J1 and J2 be two disjointly collared boundary intervals on a surface Σ, and let Σ′ be the result of gluing J1 to J2 . There is a homotopy equivalence ! hA ψ(Σ′ ) ∼ ψ(Σ) ψ

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where the action of Aψ is as follows. Given a square with ψ structure K ∈ Aψ , we glue the left edge of one copy of K to J1 and glue the left edge of a second copy of K to J2 . This proposition is the key topological input in our generalised ribbon graph. The proof is rather technical and so it will be postponed for the future paper. The idea is straightforward and we explain it now. Gluing a square at J1 has the eﬀect of simply changing the trivialization of the ψ structure at J1 , and this action is transitive in a homotopical sense, so the homotopy quotient of this action is equivalent to the space of ψ-structures on Σ that are not necessarily trivial at J1 . Thus the homotopy quotient appearing in the proposition builds a model for the space of ψ-structures on Σ such that are not necessarily trivial at J1 and J2 but are required to agree at these collars. Giving such a structure is equivalent to giving a structure on the glued surface Σ′ .

4.4

A 2-categorical model for the category of surfaces

In defining the modular operad of moduli spaces of ψ-structures, rather ! of surfaces and open embeddings, we will need than the category Surf a slightly different category Let Surf denote the topological category whose objects are collared surfaces. In rough language, a morphism Σ1 → Σ2 is a gluing of some collared boundary intervals together followed by a diffeomorphism. More precisely, the space of morphism is the disjoint union over all surfaces Σ′ obtained from Σ by gluing a number of pairs of boundary intervals together of the space of diffeomorphisms Σ′ → Σ2 . We let Discs ⊂ Surf denote the full subcategory whose objects are disjoint unions of discs each having at least 1 collared boundary interval These topological categories are difficult to work with, so it is convenient to replace them with more combinatorial models that will work just as well for our purposes. Let Surf 2 and Discs 2 denote the strict 2-categories with the same objects as Surf and Discs respectively, but with each space of diffeomorphisms replaced by the groupoid of diffeomorphisms and isotopy classes of isotopies. Proposition 4.9. Given a collared surface Σ, the nerve of the category ∼ HomSurf 2 (Σ, Σ) is weakly equivalent to the space HomSurf ! (Σ, Σ) = Diff(Σ). Proof. When X is a disc or annulus with no collared boundary components then the diffeomorphism group is homotopy equivalent to a circle.

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For any fixed diffeomorphism, there is a Z worth of isotopy classes of isotopies from it to itself, and the nerve of this Z gives the desired circle. In all other cases there is at most one isotopy class of isotopies between any two diffeomorphisms and the components of the diffeomorphism group are weakly contractible. Given a smooth sheaf ψ and a collared surface Σ, we have introduced ! the space ψ(Σ) of sections of ψ that are trivial at the collared boundary intervals. One sees that ψ! determines a continuous functor Surf → T op, which in turn determines a strict 2-functor Surf → T op 2 that we shall denote by the same symbol. In order to talk about cyclic and modular operads, we will need to versions of the above 2-categories in which the collared boundary intervals are labelled by a fixed finite set. We define a strict 2-functor S : Graphs → (Cat 2 ↓ Surf 2 ) by sending a union of corollas τ to the strict 2-category of collared surfaces with components identified with the components of τ and collared boundary intervals compatibly identified with the legs of τ . The functor to Surf 2 is given by forgetting the extra identification data. We also define D : Forests → (Cat 2 ↓ Discs 2 ) analogously. One can check that these functors S and D are actually symmetric monoidal and hence they define cyclic and modular operads respectively (albeit in somewhat odd looking ambient categories). A value S(τ ) of S consists of a 2-category C and a functor P : C → Surf . Composing the ψ! with P gives a functor which we will write (with a moderate abuse of notation) as S(τ ) → T op 2 .

4.5

The modular operad of moduli spaces

Associated with a smooth sheaf ψ and a diffeomorphism type of surfaces [Σ], there is a (homotopy theoretic) moduli space of surfaces diffeomorphic to Σ and equipped with a ψ-structure. This moduli space is simply the homotopy quotient ψ(Σ)hDiff(Σ) . If we consider collared surfaces equipped with ψ-structures that are trivial at the collared intervals then these moduli spaces collectively form

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a modular operad. However, rigorously defining this modular operad so that it is strictly associative is somewhat subtle. As a first approximation, given a finite set I, the corresponding space of the modular operad is the disjoint union ! " hDiﬀ(Σ) (1) ψ(Σ) Σ

where Σ runs over a set of representatives of diﬀeomorphism classes of surfaces equipped with a set of disjoint collared boundary intervals labelled by I. The composition maps ◦i,j and i ◦j should be induced by gluing collared boundaries. However, with this construction, the composition maps would only be associative up to A∞ homotopy. One way to resolve this issue and strictify the composition maps is to use all surfaces (within some set-theoretic universe) rather than selecting one representative from each diﬀeomorphism class. To this end we make the following definition. Definition 4.10. The moduli space modular operad Mψ associated with a connected smooth sheaf ψ is defined by sending an object τ ∈ Obj Graphs (i.e., a union of corollas) to the homotopy colimit " Mψ (τ ) = hocolim ψ. S(τ )

Similarly, we define a cyclic operad Dψ of moduli spaces of discs by " Dψ (τ ) = hocolim ψ. D(τ )

(To make sense of these symbols, please recall the abuse of notation mentioned above at the end of the previous subsection.) In light of Proposition 4.9 above, Mψ evaluated on a corolla with I legs yields a space homotopy equivalent to the homotopy quotient the the homotopy-theoretic moduli space of (1). We can now state our main theorem. Theorem 4.1. The derived modular envelope of the cyclic operad Dψ is weakly equivalent as a modular operad to Mψ .

Arc systems in a surface

In this section we shall prove that the space of decompositions of a surface into discs is contractible.

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119

Arcs and cutting

Let Σ be a collared surface with nonempty boundary. An arc in Σ is an embedding [0, 1] → Σ that sends the interior of the interval to the interior of the surface, sends the endpoints to the boundary, and meets the boundary transversally. We consider two arcs to be equivalent if they differ by reversing the direction of the interval via t "→ 1 − t. If Σ is equipped with any collars then we require that the arc is disjoint from the collars. An arc system is a finite (possibly empty) collection of disjoint arcs in Σ that divide the surface in regions homeomorphic to discs (the regions will be diffeomorphic to polygons rather than discs since they will have corners), each of which touches at least one arc or collared boundary component. Observe that a surface Σ equipped with an arc system A can by cut along the arcs of an arc system to yield a disjoint union of discs. Moreover, each of the boundary intervals created by the cutting can be collared uniquely up to diffeomorphism so that the resulting union of discs can be considered as a collared surface with one collared boundary interval for each collared boundary interval on the original surface plus one for each arc in the arc system. We shall denote this union of collared discs by ΣA . An arc system has a dual graph with one vertex for each region in the complement of the arcs, an edge for each arc, and a leg for each collar on the surface. We say that an arc system is reduced if its dual graph has the minimum possible number of bivalent vertices (1 in the case of a disc or annulus and zero in all other cases). An orientation on the surface induces a ribbon structure on the graph (i.e., a cyclic ordering of the half edges incident at each vertex).

5.2

The category of arc systems

A diﬀeomorphism of Σ sends arc systems to arc systems. An isotopy from an arc system A to an arc system B is a 1-parameter family of arc systems At such that A0 = A and A1 = B. A bijection from the arcs of A to the arcs of B is said to be admissible if it can be induced by an isotopy. Arc systems form a category A (Σ): the objects are arc systems and a morphism A → B consists an isotopy class of isotopies from A to a subsystem of B. We let A r (Σ) denote the full subcategory of reduced arc systems. There is a reduction functor R : A (Σ) → A r (Σ) that is

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defined by replacing each collection of parallel arcs with a single arc and deleting any arc that is parallel to a collared boundary interval (this is non canonical the case of a disc or annulus, but a choice can be made in order to define the functor). Theorem 5.1. The nerve of A (Σ) is contractible. Proof. The proof is divided into three cases. (1) Σ is a disc without collars. (2) Σ is an annulus without collars. (3) Σ is any other collared surface. (1) The dual graph of an arc system in the disc is a planar tree, and the set of univalent vertices (corresponding to arcs that bound discs containing no other arcs) inherits a cyclic order from the disc. Let Λ be the category of finite nonempty cyclically ordered sets and degree 1 maps (Connes’s cyclic category). There is a functor q : A → Λ given by sending an arc system to the set of univalent vertices of its dual graph. The nerve of Λ is known to be equivalent to BS 1 , and we will identify the map induced by the above functor with the map ES 1 → BS 1 . First, we show that the homotopy fibre of the map is S 1 . For any object [n] ∈ Λ, consider the over category q ↓ [n]. Let Z denote the category with a single object and a Z worth of endomorphisms. There is a functor r : Z → (q ↓ [n]) given by sending the single object to the arc system consisting of a single arc and sending the generating automorphism to the automorphism given by rotating the disc through 360 degrees. Over any object of q ↓ [n], the over category of the functor r has a nonempty set of objects and by unwinding the definitions carefully one can see that there is a unique isomorphism between any two objects. Thus the nerve of any over category of r is contractible and Quillen’s Theorem A implies that r induces a homotopy equivalence of nerves. By considering a generator of the fundamental group one can check that any morphism [n] → [m] in Λ induces a translation functor (q ↓ [n]) → (q ↓ [m]) that is a homotopy equivalence on nerves. Quillen’s Theorem B thus says that BZ → A → Λ gives a homotopy fibre sequence upon passing to nerves. The nerve of the fibre is S 1 and the nerve of the base is BS 1 . To conclude that the total space is ES 1 , we need only check that the inclusion of the fibre S 1 into the total space is trivial on π1 . The generator of π1 of the fibre is represented by a rotation of the disc through 360 degrees. Given a symmetric configuration of 3 arcs in the disc, there is an automorphism given by rotation by 120 degrees, and the cube of this automorphism is the 360 degree rotation. A calculation in

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Figure 1: Calculation in the localised category representing the total space, showing the the generator of the fundamental group of the fibre is trivial in the total space. The unlabelled arrows in this diagram are all the evident inclusions of arc systems. the localised category using §3.2 will show that this 120 degree rotation is trivial in the localisation. The calculation is represented by the diagram in figure 5.2. (2) The dual graph of an arc system in the annulus is a chain of bivalent vertices (corresponding to those arcs which go from the inner boundary to the outer one) with some trees attached (corresponding to nested sets of arcs that have both ends on the same boundary circle). This chain on bivalent vertices inherits a cyclic order from the annulus, and sending an arc system to this set defines a functor from A to Λ. Using arguments similar to case (1) above, one can conclude that the homotopy fibre of the map of nerves is S 1 and the fibre sequence is in fact S 1 → ES 1 → BS 1 . (3) In this case we use a category-theoretic reformulation of an argument from Hatcher [12]. Let A denote the category of isotopy classes of arc systems and admissible bijections. Since, in this case, every admissible bijection is induced by a unique isotopy class of isotopies, the canonical functor A → A is an equivalence of categories. Fix an arc x. We will show that the identity functor on A and

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Sx (A)

Figure 2: The eﬀect of the arc surgery operator Sx . the constant functor sending any arc system to x are homotopic after passing to nerves by constructing a zigzag of natural transformations between functors A → A , starting with the identity and finishing with the constant functor. We define an operator Sx from arcs to arcs as follows. If x and y are disjoint (up to isotopy) then Sx (y) = y. If x and y intersect then, moving them by isotopies, we may put them in a position so that they cross transversally and the number of intersection points is minimal (e.g., choose a metric and use the geodesic representatives of their isotopy classes). We may now cut y at each point where it meets x and slides the resulting endpoints along x until they reach the boundary of the surface, as shown in figure 5.2. We can extend Sx to a map from arc systems to arc systems by applying it to each of the arcs in a system. Let S1 : A → A be the functor that sends an arc system A to A ∪ Sx (A), let S2 be the functor that sends A to Sx (A), let S3 be the functor that sends A to x ∪ Sx (A), and let S4 be the constant functor sending any arc system to x. It is straightforward to see that there are natural transformations id → S1 ← S2 → S3 ← S4 induced by the evident inclusions of arc systems. Upon passing to nerves, this zigzag of natural transformation yields the desired homotopy from the identity map to the constant map. Corollary 5.1. The nerve of A r (Σ) is contractible. Proof. Let i denote the inclusion A r !→ A and observe that for any arc system A the over category i ↓ A has an initial object given by

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any choice reduction R(A) and morphism R(A) → A. The result then follows from Quillen’s Theorem A and Theorem 5.1.

Sketch of the proof Theorem 4.1

Let ψ be a smooth sheaf and recall that we have defined the cyclic operad Dψ of moduli spaces of discs, and the modular operad Mψ of moduli spaces of surfaces. We will construct a chain of weak equivalences between the derived modular envelope LMod(Dψ ) and Mψ . By the construction of the derived Kan extension along ℓ : Forests #→ Graphs (evaluated at τ as the homotopy colimit over the over category ℓ ↓ τ , we see that the derived modular envelope of Dψ , evaluated on a union of corollas γ, is given by ! hocolim Dψ = hocolim hocolim ψ. γ∈ℓ↓τ

γ∈ℓ↓τ

D(γ)

By the Fubini theorem for homotopy colimits, this is weakly equivalent ! to hocolim! D ψ. ℓ↓τ " An object of ℓ↓τ D is a graph γ (equipped with an identification π0 γ ∼ = τ ) and a decoration of each vertex by a disc. Gluing the discs together as prescribed by the graph results in a surface Σ ∈ S(τ ). Moreover, the collection of arcs in Σ along which the gluing was performed yield an arc system. This defines a strict 2-functor. Lemma " 6.1. The " above construction defines an equivalence of 2-categories ℓ↓τ D ≃ S(τ ) A . The inverse of the equivalence is constructed by cutting along the arcs, and it is denoted κ. Note that it is not a strict 2-functor but only a lax 2-functor. Hence hocolim ψ! ≃ hocolim ψ! ◦ κ, ! ! ℓ↓τ

S(τ )

which is weakly equivalent to

hocolim LR! (ψ! ◦ κ), ! S(τ )

" " where R : S(τ ) A → S(τ ) A r is the arc system reduction functor. We now come to the key step in the argument. As explained in [13], Kan extension along R can be thought of as integrating out the

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bivalent vertices in the dual graphs to the arc systems, and this has the following eﬀect. For each edge in the dual graph, one builds a 2-sided bar construction for the monoid Aψ of geometric structures on a strip acting on the the spaces associated to the vertices at either end of the edge. By Proposition 4.8 we then have the following result. " Lemma 6.2. LR! (ψ! ◦ κ) ≃ ψ! ◦ π, where π : S(τ ) A r → S(τ ) forgets the arc system.

In other words, starting with the space of all pairs of an unreduced arc system and a ψ-structure trivial on the arcs, and then integrating out the bivalent vertices in the dual graph yields a space equivalent to the space of pairs of a reduced arc system and a ψ-structure not-necessarily trivial on the arcs. " Finally, by Corollary 5.1, the homotopy colimit of ψ! ◦ π over S(τ ) A r is equivalent to the homotopy colimit of ψ! overS(τ ), which is precisely the definition of Mψ (τ ).

Acknowledgement I thank the MIMS for hosting the conference at which I first spoke about this work, and Sadok Kallel for organizing the meeting. I would also like to thank the anonymous referee for some useful comments and suggestions. Jeﬀrey Giansiracusa Department of Mathematics, Swansea University, Singleton Park, Swansea, Wales, SA2 8PP, United Kingdom, j.h.giansiracusa@swansea.ac.uk

References [1] Carrasco P.; Cegarra A. M.; Garz´on A. R., Nerves and classifying spaces for bicategories, Algebr. Geom. Topol. 10:1 (2010), 219–274. [2] Cegarra, A. M., Homotopy fiber sequences induced by 2-functors, J. Pure Appl. Algebra 215:4 (2011), 310–334. [3] Boavida de Brito P.; Weiss M., Manifold calculus and homotopy sheaves, Homology Homotopy Appl. 15:2 (2013), 361–383. [4] Ayala D., Geometric cobordism categories, arxiv:0811.2280 (2008).

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[5] Cordier J.-M.; Porter T., Homotopy coherent category theory, Trans. Amer. Math. Soc. 349:1 (1997), 1–54. [6] Gabriel P.; Zisman M., Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, SpringerVerlag New York Inc., New York, 1967. [7] Berger C.; Batanin M., Homotopy theory for algebras over polynomial monads, arXiv:1305.0086 (2013). [8] Costello K., The A–infinity operad and the moduli space of curves, arXiv:math/0402015, (2004). [9] Costello K., Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210:1 (2007), 165–214. [10] Costello K., A dual version of the ribbon graph decomposition of moduli space, Geom. Topol. 11 (2007), 1637–1652. [11] Thomason R. W., Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85:1 (1979), 91– 109. [12] Hatcher A., On triangulations of surfaces, Topology Appl. 40:2 (1991), 189–194. [13] Giansiracusa J., The framed little 2-discs operad and diﬀeomorphisms of handlebodies, J. Topol. 4:4 (2011), 919–941.

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