Morfismos, Vol 23, Num 2, 2019

VOLUMEN 23 NÚMERO 2 JULIO A DICIEMBRE 2019 ISSN: 1870-6525

VOLUMEN 23 NÚMERO 2 JULIO A DICIEMBRE DE 2019 ISSN: 1870-6525

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Morfismos

Contents - Contenido Approximating diffeomorphisms by elements of Thompson’s groups F and T Deniz E. Stiegemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A graph-theoretic viewpoint for discrete Morse theory Teresa Hoekstra Mendoza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Sequential motion planning in connected sums of real projective spaces Jesús González and Jorge Aguilar-Guzmán . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Morfismos, Vol. 23, No. 2, 2019, pp. 1–10

Morfismos,, Vol. Vol.23, 23,No. No.2,2,2019, 2019,pp. pp.1–10 1–10

Approximating diffeomorphisms by elements of Thompson’s groups F and T ∗ Approximating diffeomorphisms by elements of 1 Deniz E. Stiegemann Thompson’s groups F and T ∗ Deniz E. Stiegemann

1

Abstract We show how to approximate diffeomorphisms of the closed interval and the circle by elements of Thompson’s groups F and T , Abstract respectively. This is relevant in the context of Jones’ continuum limitWe of discrete multipartite systems and its dynamics. show how to approximate diffeomorphisms of the closed interval and the circle by elements of Thompson’s groups F and T , 2010 Mathematics Subject Classification: respectively.

This is relevant in the54C30 context of Jones’ continuum discretediffeomorphisms, multipartite systems and its dynamics.

Keywordslimit and of phrases: Thompson’s groups

1

2010 Mathematics Subject Classification: 54C30 Introduction Keywords and phrases: diffeomorphisms, Thompson’s groups

Over the past few years, V. F. R. Jones has introduced discrete ana1 ofIntroduction logues conformal field theories (CFTs) with the aim of constructing a suitable continuum limit to recover a CFT [2, 3, 4]. In the discrete Overa the pastgenerated few years,infinite V. F. group R. Jones has as introduced discrete theory, finitely known Thompson’s groupanalogues of conformal field theories (CFTs) with the aim of constructing T takes the role of Diff + (S1 ), the group of orientation-preserving diffeoa suitable continuum to recover a CFT [2, 3, the 4]. elements In the discrete morphisms of the circle. Inlimit contrast to diffeomorphisms, of theory, a finitely generated infinite group known as Thompson’s group T are piecewise-linear homeomorphisms, which explains the term ‘dis1 ), the group of orientation-preserving diffeoT takes Diff + (Sbeen crete’.

The the idearole hasofalready applied to physics in the context of morphisms of the circle. In contrast to diffeomorphisms, the elements of holography [5, 1].

T are piecewise-linear homeomorphisms, which explains theunitary term ‘disThe dynamics of the discrete theory is given by (projective) crete’. The of idea hasanalready been applied to physics in itthe representations T on appropriate Hilbert space.

While hascontext been of holography [5, 1]. ∗ The content of this article is part of the author’s doctoral [1], written The dynamics of the discrete theory is given dissertation by (projective) unitary in 2019 at Leibniz Universität Hannover under the supervision of Tobias J. Osborne.

1 representations of T on an appropriate Hilbert space. While it has been This work was supported by the DFG through SFB 1227 (DQ-mat) and the ∗ RTG 1991, ERC of grants QFTCMPS SIQS, the cluster of excellence EXC201 Thethe content this article is partand of the author’s doctoral dissertation [1], written Quantum Engineering and Space-Time Research, and the Australian Research in 2019 at Leibniz Universität Hannover under the supervision of Tobias J.CounOsborne.

1 of Excellence for Engineered Quantum Systems (EQUS, CE170100009). cil Centre This work was supported by the DFG through SFB 1227 (DQ-mat) and the RTG 1991, the ERC grants QFTCMPS and SIQS, the cluster of excellence EXC201 Quantum Engineering and Space-Time Research, and the Australian Research Coun1 cil Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009).

1

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Deniz E. Stiegemann

shown that most of these representations are topologically discontinuous and thus unphysical [3, 6], interesting exceptions may still exist. The idea – and challenge – is to find a procedure that takes a discrete theory as input and then outputs a continuous theory. Such a procedure would certainly include some kind of limit gn → f , where gn ∈ T and f ∈ Diff + (S1 ).

The purpose of this paper is to clarify how orientation-preserving diffeomorphisms of S1 can be approximated by elements of Thompsons’s group T . This includes a similar description for orientation-preserving diffeomorphisms of the interval I = [0, 1] and Thompson’s group F .

The corresponding density theorems are certainly known and have been proved for Homeo+ (I) and Homeo+ (S1 ) in a much more general setting [7, 8]. The advantage of our work is a direct proof that is hands-on for the present context and can be directly translated into an algorithm to construct approximations, suitable for the computer.

The reader who is specifically interested in computational applications can find a step-by-step outline of the construction in Section 3.1.

2

Main Facts

Recall that the dyadic rationals are all numbers of the form m/2k with m ∈ Z and k ∈ N = {0, 1, 2, . . . }. By a breakpoint of a piecewise linear function we mean the points at which it is not differentiable.

Definition 2.1. Thompson’s group F is the group of piecewise linear homeomorphisms g of the closed unit interval I = [0, 1] such that (Th1 ) the breakpoints of g and their images are dyadic rationals; (Th2 ) on intervals of differentiability, the derivatives of g are integer powers of 2; and Thompson’s group T is the group of piecewise linear homeomorphisms g of S1 with these properties.2 Let Diff 1+ (I) denote the group of orientation-preserving C 1 -diffeomorphisms of the interval, and similarly for S1 . Our result is stated in 2

These definitions of F and T differ from, but are equivalent to, the standard reference [9]. In particular, our definition of F is not minimal since it actually suffices to require that only the breakpoints are dyadic rationals. Their images are then automatically dyadic due to property (Th2 ) and the fact that 0 is a fixpoint.

Approximating Diffeomorphisms

3

terms of the C 0 -norm f = sup |f (x)|. x

Theorem 2.2. For every f ∈ Diff 1+ (I) and > 0, there exists g ∈ F such that f − g < . Similarly, if f ∈ Diff 1+ (S1 ), then there exists g ∈ T with this property.

This statement is known and follows from [8, Thm. A4.1] and [7, Prop. 4.3]. It is actually true for all orientation-preserving homeomorphisms. In Section 3 we will give a direct proof of the theorem in the present context.

The next logical question is whether there is an approximation for the first derivatives of diffeomorphisms. While generally elements of both F and T are not everywhere differentiable, we can define a function d(f, g) =

sup |f (x) − g (x)|.

x∈S1 \Bg

that measures the distance between the first derivatives of f ∈ Diff 1+ (S1 ) and g ∈ T wherever g is defined. Here Bg denotes the set of breakpoints of g. (The definition of d for Diff 1+ (I) and F is analogous.) We can therefore rephrase the question: Given a diffeomorphism f and > 0, is there a function g from the appropriate Thompson group such that d(f, g) < ? The answer is that such an approximation is not possible since the set of all integer powers of 2 is very sparse in (0, 1). This fact is made precise in the following proposition, which is similar to [10, Théorème III.2.3].

Proposition 2.3. For every f ∈ Diff 1+ (S1 ) which is not a rotation, there exists µ > 0 such that d(f, g) > µ for all g ∈ T . The same holds when S1 is replaced by I and T is replaced by F .

Here the rotations in Diff 1+ (S1 ) are all elements f with f (x) = 1 for all x ∈ S1 , which includes the identity. In Diff 1+ (I), the identity is the only rotation.

3

Approximating Diffeomorphisms

In this section, we describe the approximation procedure that represents a proof of Theorem 2.2. We begin with a few simplifying observations.

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Deniz E. Stiegemann

3/2

1

1

1/2

1/2

1/4 0∼ =1

1/2 3/4

(a)

0

1/2

1

0

(b)

1/2

1

(c)

Figure 1: Three representations of the same element of Thompson’s group T : (a) as a map S1 → S1 , here drawn by indicating how breakpoints (on the inner circle) are mapped to their images (on the outer circle); (b) the usual representation as a function [0, 1] → [0, 1]; (c) the representation as a function [0, 1] → R, which we will use – note that it is a homeomorphism onto its image [1/2, 3/2].

The graph of a piecewise linear function can be described by specifying the (finitely many) breakpoints at which the function is not differentiable, and the images of the breakpoints. For a strictly monotone piecewise linear function g, we therefore have a partition of the domain of g by points x 1 < x2 < · · · < xn and a partition of the codomain of g by the points g(x1 ) < g(x2 ) < · · · < g(xn ) such that g is the function corresponding to the curve of connected line segments through the points

x1 , g(x1 ) , x2 , g(x2 ) , . . . , xn , g(xn ) .

In the case of Thompson’s groups F and T , the breakpoints have to be at dyadic rationals.

Given any homeomorphism f : S1 → S1 , we can identify it with a homeomorphism f˜: R → R that satisfies f˜(x + 1) = f˜(x) + 1.

Approximating Diffeomorphisms

5

In particular, f˜|[0,1] is continuous, which will be needed later. An example is shown in Figure 1.

3.1

Outline of the Construction

Before we come to technical details, we present a rough outline of the proof for the case of Diff 1+ (I) and Thompson’s group F . Let f ∈ Diff 1+ (I) be given.

1. Divide the domain of f into n small intervals of equal length, where n is a power of 2. Therefore the breakpoints ξi of the partition are dyadic rationals.

2. For each breakpoint ξi choose a dyadic rational ηi close to the image f (ξi ).

3. Find a piecewise linear homeomorphism γi : [ξi , ξi+1 ] → [ηi , ηi+1 ] for each i = 0, . . . , n − 1 that serves as a dyadic interpolation from the point (ξi , ηi ) to the point (ξi+1 , ηi+1 ), which means that γi has breakpoints at dyadic rationals and its slopes are powers of 2 (Section 3.2).

By defining a function g : [0, 1] → [0, 1] whose values on the interval [ξi , ξi+1 ] are determined by γi , we obtain a homeomorphism g ∈ F close to f .

3.2

Dyadic Interpolation

Let two distinct points p = (p1 , p2 ) and q = (q1 , q2 ) in R2 be given, with p1 < q1 and p2 < q2 and such that all coordinates pi , qi are dyadic rational numbers. Then r = q − p also has dyadic rational coordinates r1 and r2 which can be written as r1 =

m1 , 2k 1

r2 =

m2 2k2

with m1 , m2 , k1 , k2 ∈ N and m1 , m2 > 0. We proceed as illustrated in Figure 2. Let (a, b) = (1, 2) if m1 ≤ m2 and (a, b) = (2, 1) if m1 > m2 , so that mb = max{m1 , m2 } and ma = min{m1 , m2 }. Set d = mb − ma .

For the moment, assume d > 0. Consider the sequence (cn ) defined by c0 = 0 and n−1

cn = ma 2i = ma (2n − 1) i=0

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Deniz E. Stiegemann

11 26

.. .

2 26 1 26

0

1 23

2 23

1 24

3 24

1 25 1 26

3 25 3 26

5 25

5 26

n=1 7 25

n=2 n=l=3

Figure 2: Illustration of how to cut the sides of a dyadic rectangle such that all sides are divided into dyadic partitions with equally many subintervals. In this example, m1 /2k1 = 11/26 and m2 /2k2 = 2/23 .

Since 11 > 2, we divide the left side of the rectangle into 11 intervals, each of length 1/26 . The bottom side is first divided into 2 intervals, each of length 1/23 . Then we successively cut all its intervals in half, repeatedly going from left to right, until the bottom side is also divided into 11 intervals. The thick line shows the graph of the piecewise linear function arising from these partitions. for n ≥ 1. Let l ≥ 1 be the smallest integer with cl ≥ d. Define a sequence of dyadic numbers ξ1 , . . . , ξd by setting ξi+cn =

2i − 1 2ka +n

for all i, n with either 1 ≤ i ≤ 2n ma and 0 ≤ n ≤ l−1, or 1 ≤ i ≤ d−cl−1 when n = l. Set

m

0 ≤ m ≤ m X=

a ∪ {ξ1 , . . . , ξd } 2k a for d > 0 and

m

0 ≤ m ≤ m

a 2ka for d = 0. We arrange the ma + d = mb elements of X in increasing order and denote them xa1 ≤ · · · ≤ xamb . They are the breakpoints of a X=

Approximating Diffeomorphisms

7

standard dyadic partition of [0, ma /2ka ] into mb intervals. Furthermore, set xbm = m/2kb for 0 ≤ m ≤ mb . The points p1 + x11 , p1 + x12 , . . . , p1 + x1n and p2 + x21 , p2 + x22 , . . . , p2 + x2n form standard dyadic partitions dividing the two intervals [p1 , p2 ] and [q1 , q2 ], respectively, into equally many subintervals. To these partitions corresponds a piecewise linear function. By construction, it is bijective, has breakpoints only at dyadic rationals, and only slopes wich are powers of 2.

3.3

Finding Dyadic Rationals

Let 0 < p < q be given. Since the dyadic rationals are dense in R, one can always find a dyadic number in the open interval (p, q). For an example, let

x + 1 if x ∈ Z, ceil(x) = min{n ∈ Z | n > x} = x otherwise.

Set

k = max 0, ceil(− log2 (q − p)) , m = ceil(2k p).

Then m, k ∈ N, and m/2k ∈ (p, q) is a dyadic rational.

3.4

The Construction

We proceed with the construction of approximations, which then proves Theorem 2.2. Let f ∈ Diff 1+ (I) and > 0 be given, and assume < 1 without loss of generality. Set S = maxx∈I f (x) and note that S ≥ 1.

∈ N and n = 2∆ , and note that ∆ ≥ 1. Set Let ∆ = − log2 3S ξi = i/n,

i = 0, . . . , n.

(This implies that ξ0 = f (ξ0 ) = 0 and ξn = f (ξn ) = 1.) Moreover, set δ = min{ /2, (f (ξn ) − f (ξn−1 )/2)}

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Deniz E. Stiegemann

and note that the interval Ii = (max{f (ξi−1 ) + δ, f (ξi )}, f (ξi ) + δ) is non-empty and a subset of (0, 1) for i = 1, . . . , n−1. We pick a dyadic rational ηi ∈ Ii for each i = 1, . . . , n − 1. Let η0 = 0 and ηn = 1, and define the function g : [0, 1] → [0, 1] by setting g(x) = γi (x)

(1)

for x ∈ [ξi , ξi+1 ] and i = 0, . . . , n − 1, where γi is a dyadic interpolation from the point (ξi , ηi ) to the point (ξi+1 , ηi+1 ). From the definitions of γ, {ξi } and {ηi } it is clear that g ∈ F . Furthermore, for all i = 0, . . . , n − 1 and x ∈ [ξi , ξi+1 ], consider the sequence of statements (2) (3) (4)

|g(x) − f (x)| ≤ g(ξi+1 ) − f (ξi )

< f (ξi+1 ) − f (ξi ) + /2 f (ξi+1 ) − f (ξi ) (ξi+1 − ξi ) + /2 = ξi+1 − ξi

(5)

< S2−∆ + /2

(6)

< /3 + /2 < .

(2) holds since f and g are strictly increasing and g(ξi ) > f (ξi ). For (3), recall that g(ξi+1 ) = ηi+1 < f (ξi+1 ) + δ. (4) to (6) are obvious. We have thus found g ∈ F with maxx∈[0,1] |f (x) − g(x)| < .

If instead f ∈ Diff 1+ (S1 ), f corresponds to a function f˜: R → R with im(f˜) = [u, u + 1] for some u ∈ R and such that f˜: [0, 1] → [u, u + 1] is a diffeomorphism (as explained above). Define S, ∆, n, ξi and Ii as above, but with δ = min{ /2, (f˜(ξ1 ) − f˜(ξ0 ))/2}. Choose ηi ∈ Ii for i = 1, . . . , n − 1 as before. Let η0 be a dyadic rational in the interval (f˜(ξ0 ) + δ, f˜(ξ1 )) and set ηn = η0 + 1. This ensures that max{f˜(ξn−1 ) + δ, f˜(ξn )} < ηn .

Now we can define a function g̃ : [0, 1] → R as in (1). It follows that (2) to (6) hold, and that g ∈ T upon taking the quotient S1 = R/Z. This concludes the proof of Theorem 2.2.

4

C 1 -Discreteness

Finally, we show that it is not possible to go beyond C 0 -approximation.

Note that the proof is also valid in the more general case when Diff 1+ (I)

9

Approximating Diffeomorphisms

and Diff 1+ (S1 ) are replaced by the sets of all differentiable bijections of I or S1 , respectively, whose inverses are also differentiable.

Proof of Proposition 2.3. Let g ∈ T and f ∈ Diff 1+ (S1 ). We will identify f and g with functions on the interval [0, 1] as before. Let x0 ∈ [0, 1]\Bg .

The two powers of 2 closest to f (x0 ) are given by 2 log2 f

(x

0 )

≤ f (x0 ) ≤ 2 log2 f

(x

0 )

.

If f (x0 ) is not a power of 2, the inequalities are strict and therefore

d(f, g) ≥ min f (x0 ) − 2 log2 f (x0 ) , f (x0 ) − 2 log2 f (x0 ) > 0.

The case that f (x0 ) is not a power of 2 for some x0 ∈ [0, 1] \ Bg occurs for all differentiable f ∈ Diff 1+ (S1 ) except for rotations. For if f is not a rotation, there exists x1 ∈ [0, 1] with f (x1 ) = c = 1. By the mean value theorem, there also exists x2 ∈ [0, 1] with f (x2 ) = 1. Without loss of generality, assume c < 1 and x1 < x2 . Then by Darboux’s theorem, [c, 1] ⊂ f ([x1 , x2 ]). Since Bg is finite, [c, 1] \ f (Bg ) ⊂ im(f ) surely contains points which are not powers of 2.

It is clear that Diff 1+ (I) and F are a special case of this argument, which concludes the proof.

Acknowledgement I would like to thank Tobias Osborne for introducing me to the problem and many helpful discussions. I am also grateful to Terry Farrelly and Ramona Wolf for numerous comments and a careful reading of the manuscript.

Deniz E. Stiegemann School of Mathematics and Physics, The University of Queensland, Brisbane Qld 4072, Australia, d.stiegemann@uq.edu.au

References [1] D. E. Stiegemann. Thompson Field Theory. Dissertation, Leibniz Universität Hannover, 2019.

[2] V. F. R. Jones. Some unitary representations of Thompsons groups F and T . J. Comb. Algebra, 1(1):1–44, 2017.

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[3] V. F. R. Jones. A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain. Commun. Math. Phys., 357(1):295– 317, 2018.

[4] V. F. R. Jones. Scale invariant transfer matrices and Hamiltionians.

J. Phys. A: Math. Theor., 51(10):104001, 2018.

[5] T. J. Osborne and D. E. Stiegemann. Dynamics for holographic codes. J. High Energy Phys., 2020(4):1–41, 2020.

[6] A. Kliesch and R. König. Continuum limits of homogeneous binary trees and the Thompson group.

[7] D. Zhuang. Irrational stable commutator length in finitely presented groups. J. Mod. Dyn., 2(3):499–507, 2008.

[8] R. Bieri and R. Strebel. On Groups of PL-homeomorphisms of the Real Line. volume 215 of Math. Surveys Monogr. Amer. Math.

Soc., 2016.

[9] J. W. Cannon, W. J. Floyd, and W. R. Parry. Introductory notes on Richard Thompson’s groups. Enseign. Math., 42:215–256, 1996.

[10] E. Ghys and V. Sergiescu. Sur un groupe remarquable de difféomorphismes du cercle. Comment. Math. Helv., 62(1):185–239, 1987.

Morfismos, Vol. 23, No. 2, 2019, pp. 11–16 Morfismos, Vol. 23, No. 2, 2019, pp. 11–16

A graph-theoretic viewpoint for discrete Morse theory A graph-theoretic viewpoint for discrete Morse Mendoza theory Teresa Hoekstra Teresa Hoekstra Mendoza Abstract A well known theorem of discrete Morse theory states that a disAbstract crete vector field is acyclic if and only if it is a gradient vector field for a A discrete Morse function f .

In this papertheory we give a simple well known theorem of discrete Morse states that a disproofcrete usingvector a wellfield known theorem in graph theory.

We the field is acyclic if and only if it is a gradientdo vector same for for aanother well known result in discrete Morse theory that discrete Morse function f . In this paper we give a simple statesproof that using in a simplicial complex endowed with atheory. discreteWe graa well known theorem in graph do the dient same vectorforfield, if two critical cells of the same dimension are that another well known result in discrete Morse theory such states that there a unique complex gradient endowed path between we grathat exists in a simplicial with them, a discrete can find a new vector field for which these two cells are not critical dient vector field, if two critical cells of the same dimension are and every critical cell aremains in path the new field. them, we such other that there exists uniquecritical gradient between can find a new vector field for which these two cells are not critical

2010 Mathematics Subject Classification: 57M15, 57Q10. and every other critical cell remains critical in the new field.

Keywords and phrases: Discrete Morse theory, gradient field, cancelling critical cells.

2010 Mathematics Subject Classification: 57M15, 57Q10.

Keywords and phrases: Discrete Morse theory, gradient field, cancelling critical cells.

1

Introduction

In this we shall give a graph-theoretic view point of two well 1 paper Introduction known theorems in discrete Morse theory.

In this shall give a graph-theoretic view point two well The first paper one is we a characterization of discrete gradient vectoroffields. theorems in proved discreteinMorse theory.

This known theorem has been [1] (page 94), where this graph theoretic view is also This ofgraph-theoretic view pointfields.

Thepoint first one is a mentioned. characterization discrete gradient vector simplifies the proof considerably and uses a well known result in graph This theorem has been proved in [1] (page 94), where this thetheory which we point also shall prove. oretic view is also mentioned. This graph-theoretic view point simplifies proof gives considerably andfor uses a well known result cells in graph The secondthe theorem a condition cancelling two critical theory which we also shall prove. in a discrete gradient vector field. It is also proved in [1] (page 110) but, to the best our knowledge, it hasanot been proved using graph-theory Theofsecond theorem gives condition for cancelling two critical cells in a discrete gradient vector field. It is also proved in [1] (page 110) but, to the best of our knowledge, it 11has not been proved using graph-theory 11

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Teresa Hoekstra Mendoza

tools. This result is very useful since critical cells play an important role in discrte Morse theory.

In both cases I will give proofs of abstract theorems in graph theory and then apply them for discrete Morse theory. I will use the usual graph-theory notation, given a digraph D, V (D) shall denote the set of vertices and A(D) the set of arrows.

2

Graph theory

Definition 2.1. Given a digraph D, x, y ∈ V (D), an xy-path is a sequence of vertices (x = x1 , x2 , ..., xn = y) such that for every i = 1, ..., n − 1 there exists an arrow (xi , xi+1 ) ∈ A(D). The length of the path is n and a cycle is a closed path in the sense that x1 = xn By an acyclic digraph we mean a digraph with no cycles. In particular an acyclic digraph has no loops and no symmetric arrows, as there would be cycles of length one and two respectively.

Lemma 2.2. Let D be an acyclic digraph and suppose γ = (x0 , ..., xn ) is the only x0 xn -path in D. If W is the digraph obtained by inverting every arrow in γ. Then W is acyclic.

Proof. Proceeding by contradiction, suppose that W has a cycle C. Let Γ be the xn x0 -path in W . Clearly C has at least one arrow in Γ. Let (xm , xm−1 , ..., xk ) be a segment of C contained in Γ with the property that neither of the two arrows (xm+1 , m) and (xk , xk−1 ) are in A(C).

Let P be the segment of C disjoint from γ that starts at the vertex xk and ends at a vertex xi for some i = 0, 1, ..., n.

1. If i < k then P ∪ (xi , xi+1 , ..., xk ) is a cycle in the digraph D, a contradiction.

2. If i > k then (x0 , x1 , ..., xk ) ∪ P ∪ (xi , xi+1 , ..., xn is another x0 xn path, a contradiction.

Hence W is acyclic.

Lemma 2.3. A finite acyclic digraph D has at least one vertex v ∈ V (D) with δ + (v) = 0 where δ + (v) = |{x ∈ V (D) : (v, x) ∈ A(D)}|.

Proof. Since D is finite and acyclic there exists a longest path in D. The last vertex of this path can not have any outward arrows.

A graph-theoretic view point for dscrete Morse theory.

13

Theorem 2.4. A finite digraph D is acyclic if and only if there exists a function f : V (D) → N ∪ {0} which decreases along directed paths.

Proof. Suppose D is acyclic. Given a vertex v, let p(v) denote the lenght of the largest path in D starting from v. Define the sets Vi = {v ∈ V (D) : p(v) = i}. Since D is finite, ni=0 Vi = V (D) for a suficiently large n. From the previous lemma we know that V0 is non empty. We define f : V (D) → N ∪ {0} given by f (x) = i for all x ∈ Vi .

We must now prove that f decreases along paths. Suppose γ is a path and let (x, y) be an arrow in γ. If p(x) ≤ p(y), denote the largest path starting from y by Py and the largest path starting from x by Px .

Then Py is longer than Px but Py ∪ (x, y) is a longer path starting from x, which is a contradiction.

If such a function f exists and {x0 , x1 , ..., xn = x0 } is a cycle then

f (x0 ) > f (x1 ) > ... > f (xn ) = f (x0 ) which is impossible.

3

Discrete Morse theory

Definition 3.1. Let X be a set and K a collection of subsets of X. We say that the pair (X, K) is a simplicial complex if τ ∈ K and ν ⊂ τ implies ν ∈ P . The elements of K are called simplexes and the dimension of a simplex τ is its cardinality minus one.

Given a simplicial complex we shall denote by σ p that the dimention of a simplex σ is p. We will denote that σ is a face of τ by σ < τ .

Definition 3.2. A discrete Morse function on a simplicial complex X is a function f : K(X) → R, where K(X) denotes the set of simplexes of X, such that given a simplex σ, |{τ ∈ K(X) : σ p > τ p−1 , f (σ) ≤ f (τ )}| ≤ 1 and |{ν ∈ K(X) : σ p < ν p+1 : f (σ) ≥ f (ν)}| ≤ 1.

A discrete Morse function can be defined on an CW-complex but for our purposes we shall only consider simplicial complexes.

Definition 3.3. A discrete vector field on a simplicial complex X is a collection of pairs of simplexes {(σ, τ ) : σ < τ, dimτ − dimσ = 1} such that every simplex is in at most one pair.

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Given a discrete Morse function f , we can obtain a discrete vector field called the gradient vector field of f .

Definition 3.4. The gradient vector field of a discrete Morse function is the vector field consisting precisely of the pairs σ p < τ p+1 for which f (τ ) ≥ f (σ).

In general we say that a discrete vector field is gradient if it is the gradient vector field of a discrete Morse function.

Definition 3.5. Given a simplicial complex X, we can associate a digraph to it called the Hasse diagram. The veritices are the simplexes of X. The set of arrows is {(τ, σ) : σ p < τ p+1 }.

When X has a discrete vector field we can indicate which pairs belong to the vector field in the Hasse diagram by inverting the corresponding arrow. We call this the modified Hasse diagram.

Definition 3.6. The simplexes that do not belong to any pair of the discrete vector field V are called the critical simplexes of V .

We shall now make an observation about the modified Hasse diagram D of a discrete vector field. When we have a gradient vector field V associated to the discrete Morse function f , notice that (α, β) ∈ A(D) if and only if |dimα − dimβ| = 1 and one of the following holds: • β > α, with f (β) ≤ f (α).

• α > β, with f (α) > f (β).

This means that a discrete Morse function does not increse along paths in the modified Hasse diagram. We shall use this observation in the following theorem.

Definition 3.7. Given a discrete vector field W , a W -path of dimention p is a sequence of p-simplexes ν1 , ν2 , ..., νk such that νi < W (νi−1 ) for i = 1, ..., k, where (νi , W (νi )) ∈ W . We say that the lenght of the path is k and that the path is closed if ν1 = νk .

Theorem 3.8. A discrete vector field W on a finite simplicial complex X is gradient if and only if it has no closed paths.

Proof. Note that in particular a closed W -path is a cycle in the modified Hasse diagram D. By theorem 2.4, the modified Hasse diagram is acyclic if and only if there exists a funcion f : X → N ∪ {0} which decreases along paths. We thus only need to show that the function we constructed

A graph-theoretic view point for dscrete Morse theory.

15

in 2.4 is indeed a discrete Morse function with gradient vector field W .

Suppose σ p−1 < τ p and ν p−1 < τ p are simplexes such that f (σ) > f (τ ) and f (ν) > f (τ ). Since f does not increase along paths, in particular it can not increase along arrows. Recalling the construction of f in theorem 2.4 we see that f decreases along arrows. This means that (σ, τ ), (ν, τ ) ∈ A(D) but this implies that τ would belong to two pairs in W which is a contradiction since W is a discrete vector field.

Similarly if τ p < αp+1 and τ p < β p+1 are simplexes such that f (τ ) ≥ f (α) and f (τ ) ≥ f (β) we reach a contradiction. Hence f is a discrete Morse function.

Consider the pairs (ν, τ ) such that | dim(ν) − dim(τ )| = 1 and one of the following holds: • ν < τ , with f (ν) ≥ f (τ ).

• τ < ν, with f (ν) ≤ f (τ ).

Note that these are precisely the pairs of W and therefore f has discrete gradient vector field W .

Theorem 3.9. Let V be a discrete gradient vector field. Let α and β be two critical simplexes such that dimα = dimβ − 1. Suppose there exists a unique path from β to α in the modified Hasse diagram. Then there exists a discrete gradient vector field W on X for which the set of critical simplexes is: {τ ∈ X − {α, β} : τ is critical for V }.

Proof. Let γ be the unique path from β to α in the modified Hasse diagram G of V . We shall define W by constructing its modified Hasse diagram D. Let D be the digraph obtained from G by reversing every arrow in γ. From lemma 2.2 we know that no cycles are created in D. This means that W is also a discrete gradient vector field. Now let us look at the critical simplexes of W . Notice that every simplex outside of γ is critical for W if and only if it is critical for V . For the simplexes in γ different from α and β, which were all non-critical for V , they remain non-critical for W . As for α and β, since in D one of their incident arrows has been reversed they are not critical for W .

Teresa Hoekstra Mendoza Department of Mathematics, Cinvestav Av. Instituto Politécnico Nacional 2508 Col. San Pedro Zacatenco México, D.F. CP 07360. idskjen@math.cinvestavmx

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References [1] Kevin P. Knudson. Morse theory: Smooth and discrete. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

Morfismos, Vol. 23, No. 2, 2019, pp. 17–21 Morfismos, Vol. 23, No. 2, 2019, pp. 17–21

Sequential motion planning in connected sums of real projective spaces Sequential motion planning in connected sums of real projectiveJesús spaces Jorge Aguilar-Guzmán González Jorge Aguilar-Guzmán

Jesús González

Abstract In this short note we observe that the higher topological complexity of an iterated connectedAbstract sum of real projective spaces is maximal possible.

Unlike the case of regular TC, the result iscomIn this short note we observe that the higher topological accessible through easy mod 2 zero-divisor cup-length consideraplexity of an iterated connected sum of real projective spaces is tions.maximal possible. Unlike the case of regular TC, the result is accessible through easy mod 2 zero-divisor cup-length considera-

2010 Mathematics Subject Classification: Primary 55S40, 55M30; Sections. ondary 70Q05.

Keywords and phrases:Subject higherClassification: topological complexity, connected sum,Sec2010 Mathematics Primary 55S40, 55M30; real ondary projective space.

70Q05.

Keywords and phrases: higher topological complexity, connected sum, real projective space.

1

Introduction

It was in [7] that the topological complexity (TC) of the m-th 1 proved Introduction dimensional real projective space RPm agrees1 with Imm(RPm ), the admits a smooth immersion Rdm-th . minimal dimension so that that the RPmtopological It was proved ind [7] complexity (TC) ofinthe m m 1 Cohen and Vandembroucq have recently shown in [4] that fact above agrees withthe Imm(RP ), the dimensional real projective space RP thethat g-iterated connected sum immersion of RPm with doesminimal not holddimension for gRPmd, so a smooth in Rd .

RPm admits m itself, if g ≥and 2. Indeed, TC(gRPhave ) isrecently maximal possible g ≥ 2, Cohen Vandembroucq shown in [4]whenever that the fact above m m a result currently openconnected problem ofsum assessing , the g-iterated of RPhow with doesthat notcontrasts hold for with gRP the m m ) deviates from 2m. much TC(RP itself, if g ≥ 2. Indeed, TC(gRP ) is maximal possible whenever g ≥ 2, m Cohen result for TC(gRP ) extends their im-how a resultand thatVandembroucq’s contrasts with the currently open problem of assessing m pressive [5], using theory, of the topological deviates fromobstruction 2m. muchcalculation TC(RP ) in m complexity of non closed result surfaces. this short note wetheir ob- imCohen and orientable Vandembroucq’s for In TC(gRP ) extends servepressive that a calculation simple minded cup-length argument in [5],zero-divisor using obstruction theory, of the suffices topological surfaces. In this short note we ob1 complexity of non orientable closed m This characterization holds as long as RP is not parallelizable; for the three m m serve that a simple minded zero-divisor argument suffices exceptional cases the relation is TC(RP ) = Imm(RPcup-length ) − 1 = m for m = 1, 3, 7.

1

This characterization holds as long as RPm is not parallelizable; for the three exceptional cases the relation is TC(RP 17 m ) = Imm(RPm ) − 1 = m for m = 1, 3, 7.

17

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Jorge Aguilar-Guzmán and Jesús González

to prove the analogous fact for Rudyak’s higher topological complexity TCs : Theorem 1.1. For g, m ≥ 2 and s ≥ 3, TCs (gRPm ) = sm.

This is the same (but much simplified) phenomenon for TCs (RPm ) studied in [3, 6]. The case m = 2 is essentially contained in [8, Proposition 5.1].

Remark 1.2. Since (higher) topological complexity is a homotopy invariant of spaces, Theorem 1.1 describes the corresponding invariant for any space in the homotopy type class of an iterated connected sum of a real projective space. This covers, for instance, manifolds classified up to homeomorphism in [2] (the case g = 2 in Theorem 1.1).

2

Proof

We assume familiarity with the basic ideas, definitions and results on Rudyak’s higher topological complexity, a variant of Farber’s original concept (see [1]). In what follows all cohomology groups are taken with mod 2 coefficients.

The first ingredient we need is the well-known description of the cohomology ring of the connected sum M #N of two n-manifolds M and N : Using the cofiber sequence S n−1 → M #N → M ∨ N one can see that the cohomology ring H ∗ (M #N ) is the quotient of H ∗ (M ∨ N ) by the ideal generated by the sum [M ]∗ + [N ]∗ of the duals of the (mod 2) fundamental classes of M and N . In particular, for the g-iterated connected sum gRPm of RPm with itself, we have: Lemma 2.1. The cohomology ring of gRPm is generated by 1-dimensional cohomology classes xu , for 1 ≤ u ≤ g, subject to the three relations: • xu xv = 0, for u = v; • xm+1 = 0; u m • xm u = xv .

xm u

The top class in H ∗ (gRPm ) is denoted by t; it is given by any power with 1 ≤ u ≤ g.

The higher topological complexity of gRPm

19

Corollary 2.2. The cohomology ring of the s-fold cartesian product of gRPm with itself is given by (1)

H (gRP × · · · × gRP ) ∼ = ∗

m

m

s

j=1

Z2 [x1,j , . . . , xg,j ]/Ig,j .

Here xu,j is the pull back of xu ∈ H 1 (RPm ) under the j-projection map (RPm )×s → RPm , and Ig,j is the ideal generated by the elements xm+1 u,j , m m xu,j + xv,j and xu,j xv,j for u = v.

We let tj ∈ H m ((RPm )×s ) stand for the image of the top class t ∈ H m (RPm ) under the j-th projection map (RPm )×s → RPm . The top class in (1) is then the product t1 t2 · · · ts , which agrees with any m m product xm u1 ,1 xu2 ,2 · · · xus ,s .

The second ingredient we need concerns with standard estimates for the higher topological complexity of CW complexes: Lemma 2.3 ([1, Theorem 3.9]). For a path connected CW complex X, zcls (X) ≤ TCs (X) ≤ s dim(X),

where zcls (X) is the maximal length of non-zero cup products of s-th zero divisors, i.e., of elements in the kernel of the s-iterated cup-product map H ∗ (X)⊗s → H ∗ (X).

Note that any element xr,i +xr,j is a zero-divisor, so that Theorem 1.1 follows from: Proposition 2.4. The product (x1,1 + x1,2 )m (x1,1 + x1,3 )m · · · (x1,1 + x1,s )m (x2,1 + x2,2 )m−1 (x2,1 + x2,3 ) is the top class in H ∗ ((gRPm )⊗s ) provided g, m ≥ 2 and s ≥ 3.

Proof. The case s = 3 follows from a direct calculation: (x1,1 + x1,2 )m (x1,1 + x1,3 )m (x2,1 + x2,2 )m−1 (x2,1 + x2,3 ) m m−1

m

m−i m−i m−i−1 i i i x2,1 + x2,3 x1,1 x1,2 x1,1 x1,3 x2,1 x2,2 = i=0

i=0

i=0

m−1 m−1 m m = (xm 1,1 + · · · + x1,2 )x1,3 (x2,1 + · · · + x2,2 )(x2,1 + x2,3 )

m−1 m−1 m m = (xm 1,1 + · · · + x1,2 )x1,3 (x2,1 + · · · + x2,2 )x2,1

m−1 m−1 m = xm 1,2 x1,3 (x2,1 + · · · + x2,2 )x2,1

m m−1 m m m = xm 1,2 x1,3 x2,1 x2,1 = x1,2 x1,3 x2,1 = t1 t2 t3 .

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Jorge Aguilar-Guzmán and Jesús González

Note that the second equality above holds because of the description of the ideal Ig,s : the factor t3 in the top class t1 t2 t3 can only arise from the summand xm 1,3 in the second factor of the product on the right of the first equality above. Likewise, the third equality above comes from the relation x1,3 x2,3 = 0, the fourth equality above comes from the relation x1,1 x2,1 = 0, and the fifth equality above comes from the relation x1,2 x2,2 = 0. The general case then follows easily from induction: (x1,1 +x1,2 )m (x1,1 +x1,3 )m · · · (x1,1 +x1,s+1 )m (x2,1 +x2,2 )m−1 (x2,1 +x2,3 ) = t1 · · · ts (x1,1 + x1,s+1 )m = t1 · · · ts xm 1,s+1 = t1 · · · ts+1 ,

where the next-to-last equality holds because xm+1 1,1 = 0.

Jorge Aguilar-Guzmán Departamento de Matemáticas, Cinvestav del I.P.N., Av. I.P.N. # 2508, México City 07000, México, jaguzman@math.cinvestav.mx

Jesús González Departamento de Matemáticas, Cinvestav del I.P.N., Av. I.P.N. # 2508, México City 07000, México, jesus@math.cinvestav.mx

References [1] Ibai Basabe, Jesús González, Yuli B. Rudyak, and Dai Tamaki.

Higher topological complexity and its symmetrization. Algebr.

Geom. Topol., 14(4):2103–2124, 2014.

[2] Jeremy Brookman, James F. Davis, and Qayum Khan. Manifolds homotopy equivalent to P n #P n . Math. Ann., 338(4):947–962, 2007.

[3] Natalia Cadavid-Aguilar, Jesús González, Darwin Gutiérrez, Aldo Guzmán-Sáenz, and Adriana Lara. Sequential motion planning algorithms in real projective spaces: an approach to their immersion dimension. Forum Math., 30(2):397–417, 2018.

[4] Daniel C. Cohen and Lucile Vandembroucq. Motion planning in connected sums of real projective spaces. arXiv:1807.09947 [math.AT].

[5] Daniel C. Cohen and Lucile Vandembroucq. Topological complexity of the Klein bottle. Journal of Applied and Computational Topology, 1(2):199–213, 2017.

The higher topological complexity of gRPm

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[6] Donald M. Davis. A lower bound for higher topological complexity of real projective space. J. Pure Appl. Algebra, 222(10):2881–2887, 2018.

[7] Michael Farber, Serge Tabachnikov, and Sergey Yuzvinsky. Topological robotics: motion planning in projective spaces. Int. Math.

Res. Not., (34):1853–1870, 2003.

[8] Jesús González, Bárbara Gutiérrez, Darwin Gutiérrez, and Adriana Lara. Motion planning in real flag manifolds. Homology Homotopy Appl., 18(2):359–275, 2016.

Morfismos se imprime en el taller de reproducción del Departamento de Matemáticas del Cinvestav, localizado en Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, México, D.F. Este número se terminó de imprimir en el mes de diciembre de 2020. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo técnico: Omar Hernández Orozco.