Morfismos, Vol 17, No 1, 2013

VOLUMEN 17 NÚMERO 1 ENERO A JUNIO DE 2013 ISSN: 1870-6525

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VOLUMEN 17 NÚMERO 1 ENERO A JUNIO DE 2013 ISSN: 1870-6525

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Morfismos

Contents - Contenido Partial monoids and Dold-Thom functors Jacob Mostovoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

´ Algebra C ∗ generada por operadores de Toeplitz con s´ımbolos discontinuos en el espacio de Bergman arm´ onico Maribel Loaiza and Carmen Lozano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Ideals, varieties, stability, colorings and combinatorial designs Javier Mu˜ noz, Feli´ u Sagols, and Charles J. Colbourn . . . . . . . . . . . . . . . . . . . . 41

Morfismos, Vol. 17, No. 1, 2013, pp. 1–18

Partial monoids and Dold-Thom functors Jacob Mostovoy

∗

1

Abstract Dold-Thom functors generalize infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of Γ-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting a partial monoid corresponding to a spectrum.

2000 Mathematics Subject Classification: Primary 55N20, Secondary 22A30.

Keywords and phrases: Dold-Thom functors, symmetric products, generalized homology.

1

Introduction

Let SP ∞ X be the infinite symmetric product of a pointed connected cell complex X. Then, according to the Dold-Thom Theorem [4], the homotopy groups of SP ∞ X coincide, as a functor, with the reduced singular homology of X. Although there is no computational advantage in this definition of singular homology, it is important since it can be extended to the algebro-geometric context; in particular, the motivic cohomology of [9] is defined in this way.

The construction of the infinite symmetric product can be generalized so as to produce an arbitrary connective homology theory. Such generalized symmetric products were defined by G. Segal in [11]: essentially, these are labelled configuration spaces, with labels in a Γ-space ∗ 1

Invited paper.

This work was partially supported by the CONACyT grant no. 44100.

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Jacob Mostovoy

(see also [2, 3, 6]). If the Γ-space of labels is injective (see [13]) it gives rise to a partial abelian monoid; it has been proved that in [13] that for each connective homology theory there exists an injective Γ-space. In this case Segal’s generalized symmetric product can be thought of as a space of configurations of points labelled by composable elements of a partial monoid. An explicit example discussed in [12] is the space of configurations of points labelled by orthogonal vector spaces: it produces connective K-theory.

We shall call the generalized symmetric product functor with points having labels, or “multiplicities”, in a partial monoid M , the DoldThom functor with coefficients in M . Certainly, the construction of such functors via Γ-spaces is most appealing. However, if we start with a spectrum, constructing the corresponding Dold-Thom functor using Γ-spaces is not an entirely straightforward procedure since the Γ-space naturally associated to a spectrum is not injective. The purpose of the present note is to show how a connective spectrum M gives rise to an explicit partial monoid M such that the homotopy of the DoldThom functor with coefficients in M coincides, as a functor, with the homology with coefficients in M. The construction is based on a trivial observation: if Y is a space and X is a pointed space, the space of maps from Y to X has a commutative partial multiplication with a unit.

2

Partial monoids and infinite loop spaces

2.1

Partial monoids.

Most of the following definitions appear in [10]. A partial monoid M is a topological space equipped with a subspace M(2) ⊆ M × M and an addition map M(2) → M , written as (m1 , m2 ) → m1 + m2 , and satisfying the following two conditions: • there exists 0 ∈ M such that 0 + m and m + 0 are defined for all m ∈ M and such that 0 + m = m + 0 = m; • for all m1 , m2 and m3 the sum m1 +(m2 +m3 ) is defined whenever (m1 + m2 ) + m3 is defined, and both are equal.

We shall say that a partial monoid is abelian if for all m1 and m2 such that m1 + m2 is defined, m2 + m1 is also defined, and both expressions are equal.

Partial monoids and Dold-Thom functors

3

The classifying space BM of a partial monoid M is defined as follows. Let M(k) be the subspace of M k consisting of composable k-tuples. The M(k) form a simplicial space, with the face operators ∂i : M(k) → M(k−1) and the degeneracy operators si : M(k) → M(k+1) defined as if i = 0 ∂i (m1 , . . . , mk ) = (m2 , . . . , mk ) = (m1 , . . . , mi + mi+1 , . . . , mk ) if 0 < i < k if i = k = (m1 , . . . , mk−1 ) and si (m1 , . . . , mk ) = (m1 , . . . , mi , 0, mi+1 , . . . , mk )

if 0 ≤ i ≤ k.

The classifying space BM is the realization of this simplicial space. In the case when M is a monoid, BM is its usual classifying space. If M is a partial monoid with a trivial multiplication (that is, the only composable pairs of elements in M are those containing 0), the space BM coincides with the reduced suspension ΣM .

A homomorphism f : M → N is a map such that whenever m1 + m2 is defined, f (m1 ) + f (m2 ) is also defined and equal to f (m1 + m2 ). If f , considered as a map of sets, is an inclusion, we say that M is a partial submonoid of N .

2.2

Partial monoids and spectra.

Given a pointed space X we shall write ΩX for the space all maps R → X supported (that is, attaining a value distinct from the base point of X) inside a compact subset of R. If Ω X denotes the space of all maps R → X supported in [− , ], with the compact-open topology, ΩX is given the weak topology of the union ∪ Ω X. The usual loop space can be identified with Ω1 X and the inclusion Ω1 X → ΩX is a homotopy equivalence.

If X is an abelian partial monoid and Y is a space, the space of all continuous maps Y → X is also an abelian partial monoid. Two maps f, g are composable in this partial monoid if at each point of Y their values are composable. In particular, if X is a pointed topological space, then, considering X as a monoid with the trivial multiplication, we see that ΩX is an abelian partial monoid; two maps in ΩX are composable if their supports are disjoint.

The space ΩX as defined here is much better behaved with respect to this partial multiplication than the usual loop space: while a generic

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Jacob Mostovoy

element of the usual loop space is only composable with the base point, each element of ΩX is composable with a big (in a sense that we need not make precise here) subset of ΩX. The partial multiplication on Ωn X for n > 1 can be defined inductively for all the pairs of maps from R to the partial monoid Ωn−1 X whose values are composable at each point. This is, of course, the same as treating the points of Ωn X as maps Rn → X and defining the composable pairs of maps as those with disjoint supports in Rn .

Let now M be a connective spectrum. We construct a partial abelian multiplication on its infinite loop space as follows. First, let us replace inductively M0 by a point and the spaces Mi for i > 0 by the mapping cylinders of the structure maps ΣMi−1 → Mi , obtaining a spectrum M . This, in particular, allows us to assume that all the structure maps are inclusions. As a consequence, we have inclusion maps Ωi−1 M i−1 → Ωi M i , which send composable k-tuples of elements to composable k-tuples for all k. The union of all the spaces M [i] = Ωi M i is the infinite loop space for the spectrum M and it naturally has the structure of a partial abelian monoid.

2.3

Dold-Thom functors.

Let M be an abelian partial moniod and X a topological space with the base point ∗. We define the configuration space Mn [X] of at most n points in X with labels in M as follows. For n > 0 let Wn be the subspace of the symmetric product SP n (X ∧ M ) consisting of points

n i=1 (xi , mi ) such that the mi are composable; W0 is defined to be a point. The space Mn [X] is the quotient of Wn by the relations (x, m1 ) + (x, m2 ) + . . . = (x, m1 + m2 ) + . . . , where the omitted terms on both sides are understood to coincide, and (x, 0) = (∗, m)

Partial monoids and Dold-Thom functors

5

for all x and m. This quotient map commutes with the inclusions of Wn into Wn+1 coming, in turn, from the inclusions SP n (X ∧ M ) → SP n+1 (X ∧ M ), and, hence, Mn [X] is a subspace of Mn+1 [X].

The Dold-Thom functor of X with coefficients in M is the space

Mn [X].

M [X] = n>0

The Dold-Thom functor with coefficients in the monoid of nonnegative integers is the infinite symmetric product. If M has trivial multiplication, we have M [X] = M1 [X] = X ∧ M.

The composability of labels in a configuration is essential for the functoriality of M [X]. A based map f : X → Y induces a map M [f ] : M [X] → M [Y ] as follows: a point (xi , mi ) is sent to the point (yj , nj ) where the label nj is equal to the sum of all the mi such that f (xi ) = yj .

Apart from the infinite symmetric products, Dold-Thom functors generalize classifying spaces: for any partial monoid M its classifying space BM is homeomorphic to M [S 1 ]. To construct the homeomorphism, take the lengths of the intervals between the particles to be the barycentric coordinates in the simplex in BM defined by the labels of the particles. Similarly, the classifying space of an arbitrary Γ-space can be constructed in this way (modulo some technical details), see Section 3 of [11]. The identification of BM with M [S 1 ] also makes sense for nonabelian monoids. The construction of a classifying space for a monoid as a space of particles on S 1 was first described in [8].

In the case when M is a partial abelian monoid coming from a spectrum M it is convenient to consider a different topology on Mn [X]. [i] Namely, we define Mn [X] as the the union of the spaces Mn [X] with the weak topology. Then, as above, the Dold-Thom functor with coefficients in M associates to a space X the space M [X] = ∪ Mn [X]. The technical advantage provided by this modification is that for any compact space [i] Y the map Y → M [X] factorizes through Mn [X] for some i and n.

We have the following generalization of the Dold-Thom theorem:

Theorem 2.3.1. Let M be partial abelian monoid corresponding to a connective spectrum M. Then the spaces M [S n ] form a connective spectrum weakly equivalent to M. The functor X → π∗ M [X]

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Jacob Mostovoy

coincides with the reduced homology with coefficients in M.

2.4

Other partial monoids

The subject of this note are the partial monoids coming from spectra.

Nevertheless, it is worth pointing out that there are many examples of partial abelian monoids, apart from infinite loop spaces, such that the corresponding Dold-Thom functors are linear. It is not hard to prove using the methods similar to those of Dold and Thom that if a partial monoid M has a good base point and if the complement of the subspace of composable pairs has, in a certain sense, infinite codimension in M × M , then the corresponding Dold-Thom functor defines a homology theory. We have already mentioned Segal’s partial monoid of vector subspaces of R∞ defined in [12]. Among other examples we have: 1. The configuration space of several (between 0 and ∞) distinct points in R∞ , with the sum of two disjoint configurations being defined as their union. For configurations with points in common the sum is not defined. The unit is the point ∅ thought of as the configuration space of 0 points. More generally, one can consider the configuration spaces of distinct particles in R∞ , labelled by points of a fixed space M . This was done in the paper [14] by K. Shimakawa; the homology theory produced by this construction assigns to X the stable homotopy of X ∧ M . (In fact, Shimakawa considers a more general situation of configurations in R∞ with partially summable labels belonging to some partial abelian monoid. Such configuration space is then itself a partial abelian monoid which, as Shimakawa proves, always gives rise to a homology theory.)

2. The space of all n-dimensional closed compact submanifolds of R∞ with the sum being the union of the submanifolds, whenever they do not intersect. This construction, however, adds nothing substantially new to the previous example. Indeed, since the dimension of submanifolds is finite and their codimension is infinite, their connected components can be shrunk in size simultaneously (at least in a compact family of submanifolds) and we see that the partial monoid of n-submanifolds of R∞ is weakly homotopy equivalent to the labelled configuration space of particles in R∞ , with labels in ∪M BDiff(M ), the space of all connected n-submanifolds of R∞ .

3. The space of all (piecewise smooth) spheres in R∞ . The operation is the join of two spheres inside R∞ , and it is defined whenever any two intervals connecting points of the two summands are disjoint.

Partial monoids and Dold-Thom functors

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7

Properties of Dold-Thom functors coming from spectra

Given a partial abelian monoid M , a subset Z ⊂ M , and a homotopy st : M → M, with t ∈ [0, 1], we say that st is a deformation of M , admissible with respect to Z if • s0 = Id and st (0) = 0 for all t; • for any set of composable elements mi ∈ M , the set st (mi ) is also composable for all t; • if a set of composable elements mi ∈ M is composable with m ∈ Z, then the set st (mi ) is composable with m for all t.

In what follows M = ∪i M [i] is a partial abelian monoid coming from a connective spectrum.

Lemma 3.1. For each i and for any compact subset Z ⊂ M [i] there exists a deformation dit : M [i] × [0, 1] → M [i+1] , admissible with respect to Z, such that any element of di1 (M [i] ) is composable in M [i+1] with any element of Z. Proof. Consider the points of Z and those of M [i] as maps of Ri+1 to M i+1 ; let x1 , . . . , xi+1 be the coordinates in Ri+1 . Since Z is compact there exists a ∈ R such that f ∈ Z implies that the support of f is contained in the half-space xi+1 < a.

Now, define dit for 0 ≤ t ≤ 1 by letting dit (f )(x1 , . . . , xi+1 ) to be

1 , if xi+1 > a + 1 − t−1 ; f x1 , . . . , xi , xi+1 − xi+1 − a − 1 + t−1 base point in M i+1 , if xi+1 ≤ a + 1 − t−1 .

The support of each element of di1 (M [i] ) is contained in the half-space xi+1 > a, therefore each element of di1 (M [i] ) is composable with each element of Z in M [i+1] . Also, the deformation dit , as a deformation of M [i] , is admissible with respect to Z. Indeed, dit does not change the support of f , and the composability of two elements of M [i] only depends on whether their supports are disjoint or not.

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Jacob Mostovoy

Lemma 3.2. Let I be the abelian monoid whose elements are points of [0, 1] with the sum of two numbers being their maximum. For each i there is homomorphism of partial monoids hi : M [i] → I which only vanishes at zero, and a deformation ui : M [i] × [0, 1] → M [i] , which is constant on the set hi = 1, decreases the value of hi strictly monotonically on the set hi < 1, at the value of the parameter t < 1 is a homeomorphism and at t = 1 retracts the subspace hi < 1/4 into the base point. Moreover ui and the restriction of ui+1 to M [i] are homotopic.

Proof. First we define inductively a collection of functions li : M i → [0, 1]. The space M 0 is a point and we set li to be equal to zero on it.

Assume that we have already defined the function lk . By construction, the space M k+1 is the mapping cylinder of the map ΣM k → Mk+1 induced by the structure map of the spectrum M. Consider the reduced suspension ΣM k as the product M k × [−1, 1] with the identifications (∗, s) ∼ (x, 1) ∼ (x, −1) for all x ∈ M k and s ∈ [−1, 1], and let τ be the cylinder coordinate, which is equal to 0 on ΣM k and to 1 on Mk+1 . Then we set lk+1 = 1 if τ = 1 and lk+1 ((x, s), τ ) = max (min (lk (x), 2 − 2|s|), τ ) for (x, s) ∈ ΣM k and τ < 1.

In the same vein, define a collection of retractions wi : M i × [0, 1] →

Mi . Let qt with t ∈ [0, 1] be a continuous family of continuous monotonic functions from [0, 1] to itself such that • qt (0) = 0 and qt (a) = a for all t and a ≥ 1/2; • qt is strictly monotonic for t < 1, and q1 (a) = 0 for a < 1/4; • qt (a) > qt (a) for all 0 < a < 1/2 and t < t .

Partial monoids and Dold-Thom functors

Also define r(Ď&#x201E;, l) as   0     2Ď&#x201E; â&#x2C6;&#x2019; 1 r(Ď&#x201E;, l) =  2Ď&#x201E; l â&#x2C6;&#x2019; Ď&#x201E;     â&#x2C6;&#x2019;2Ď&#x201E; l + 3Ď&#x201E; + 2l â&#x2C6;&#x2019; 2

if if if if

0 â&#x2030;¤ Ď&#x201E; â&#x2030;¤ 1/2, 1/2 < Ď&#x201E; â&#x2030;¤ 1, 0 â&#x2030;¤ Ď&#x201E; â&#x2030;¤ 1/2, 1/2 < Ď&#x201E; â&#x2030;¤ 1,

9

0 â&#x2030;¤ l â&#x2030;¤ 1/2; 0 â&#x2030;¤ l â&#x2030;¤ 1/2; 1/2 < l â&#x2030;¤ 1; 1/2 < l â&#x2030;¤ 1.

Then, assuming that we have already defined wk , we set wk+1 to be constant on Mk+1 and on the set Ď&#x201E; < 1 we define

(wk+1 )t ((x, s), Ď&#x201E; ) = ((wk )t (x), 1 â&#x2C6;&#x2019; qt (1 â&#x2C6;&#x2019; s)), tr(Ď&#x201E;, l) + (1 â&#x2C6;&#x2019; t)Ď&#x201E;

if s â&#x2030;Ľ 0, and

(wk+1 )t ((x, s), Ď&#x201E; ) = ((wk )t (x), â&#x2C6;&#x2019;1 + qt (1 + s)), tr(Ď&#x201E;, l) + (1 â&#x2C6;&#x2019; t)Ď&#x201E;

for negative s.

It is verified directly that wi is constant on the set li = 1, decreases the value of li strictly monotonically on the set li < 1, at the value of the parameter t < 1 is a homeomorphism and at t = 1 deforms the subspace li < 1/4 into the base point. Now, define the function hi on M [i] on a map Îą : Ri â&#x2020;&#x2019; M i as the maximal value of hi â&#x2014;Ś Îą and the deformation ui of M [i] as that induced by the deformation wi . It is clear that the conditions of the lemma are then satisfied.

Lemma 3.3. The set Ď&#x20AC;0 (M ) is an abelian group with the addition induced by the partial addition in M .

Proof. It is sufficient to notice that if points of M are thought of as maps of R to â&#x201E;Śiâ&#x2C6;&#x2019;1 M i for some i, an inverse to a map Îł in Ď&#x20AC;0 M will be given by Îł(C â&#x2C6;&#x2019; t) for sufficiently big C. For a space X let M<a> [X] â&#x160;&#x201A; M [X] be the subset of configurations whose coefficients are composable with a â&#x2C6;&#x2C6; M .

Lemma 3.4. The inclusion of M<a> [X] into M [X] induces isomorphisms on all homotopy groups. Moreover, if (xi , mi ) â&#x2C6;&#x2C6; M<a> [X], the map M<a+ mi > [X] â&#x2020;&#x2019; M<a> [X]

given by summing with (xi , mi ), is also a weak homotopy equivalence.

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Proof. We first need to show that the image of any map f of a finite cell complex Y into M [X] can be deformed into M<a> [X]. Assume that the image of f is contained in M [i] [X]. Then, applying the deformation dit from Lemma 3.1, with Z = {a}, to all the labels of the configurations f (y), where y ∈ Y , we get a homotopy of f to a map of Y into M<a> [X].

In order to establish the second claim, for each mi choose m ˜ i in such a way that the ˜ i is inverse to that of mi in π0 M and so that class of m the sum a + mi + m ˜ i is defined.

It is then easy to see that adding (xi , mi ) and then (xi , m ˜ i) to a configuration in M<a+ mi + m ˜ i > [X] is homotopic to the natural inclusion M<a+ mi + m ˜ i > [X] → M<a> [X].

Since all the natural inclusions between the spaces M<a> [X] for different a are homotopy equivalences, it follows that the map in the statement of the lemma is surjective on the homotopy groups. Replacing in this

argument a by a + mi we see that the map

M<a+ mi + m ˜ i > [X] → M<a+ mi > [X],

˜ i ), is also surjective on homotopy and this given by adding (xi , m proves the lemma.

4

Quasifibrations of Dold-Thom functors

The proof of Theorem 2.3.1 is based on the original argument of Dold and Thom [4], see also [1, 5]. We have the following fact: Proposition 4.1. Let M be a partial monoid coming from a connective spectrum. If X is a cell complex and A ⊂ X is a subcomplex, the map M [X] → M [X/A] is a quasifibration with the fibre M [A].

The rest of this section is dedicated to the proof of this statement.

Note that we do not require A to be connected.

We shall need the following criterion for quasifibrations. Let p : E → B be a map which is quasifibration over B ⊂ B. Assume that for any compact C ⊂ B there is a homotopy ft of the inclusion map i : C → B to a map C → B , which maps C ∩B to B for all t. Further, ⊆ p−1 (C) there is a homotopy f˜t of the suppose that for any compact C

Partial monoids and Dold-Thom functors

11

∩ p−1 (B ) → E to a map C → p−1 (B ) which maps C inclusion map C −1

to p (B ) for all t, and such that p ◦ f˜t = ft ◦ p.

Moreover, assume that ft and f˜t are well-defined up to homotopy.

Take a point b ∈ B and ˜b ∈ p−1 (b). Then we have two paths: from b to b ∈ B and from ˜b ∈ p−1 (b) to ˜b ∈ p−1 (b ), the latter covering the former. This gives a map of the homotopy groups πi (p−1 (b), ˜b) → πi (p−1 (b ), ˜b ).

Lemma 4.2. Assume that all the above maps of homotopy groups of the fibres are isomorphisms for all i ≥ 0. Then the map p is a quasifibration.

This lemma is a version of Hilfssatz 2.10 of [4] and the proof is, essentially, the same.

We denote by p the projection map X → X/A and by π the induced map M [X] → M [X/A].

We shall prove by induction on n that the map π is a quasifibration over Mn [X/A]. This, by Satz 2.15 of [4] (or Theorem A.1.17 of [1]) will imply that π is a quasifibration over the whole M [X/A].

Assume that π is a quasifibration over Mn−1 [X/A]. According to Satz 2.2 of [4] (or Theorem A.1.2 of [1]) it is sufficient to prove that π is a quasifibration over Mn [X/A] − Mn−1 [X/A], over a neighbourhood of Mn−1 [X/A] in Mn [X/A] and over the intersection of this neighbourhood with Mn [X/A] − Mn−1 [X/A].

It will be convenient to speak of delayed homotopies. A delayed homotopy is a map ft : A × [0, 1] → B such that for some ε > 0 we have ft = f0 when t ≤ ε. A map p : E → B is said to have the delayed homotopy lifting property if it has the homotopy lifting property with respect to all delayed homotopies of finite cell complexes into B. It is clear that a map that has the delayed homotopy lifting property is a quasifibration.

Lemma 4.3. Let B be an arbitrary subspace of Mn [X/A]−Mn−1 [X/A].

The map π, when restricted to π −1 (B), has the delayed homotopy lifting property.

12

Jacob Mostovoy

Proof. Let ft : Z × [0, 1] → B be a delayed homotopy of a finite cell complex Z into B, such that ft = f0 for t ≤ ε, and let f˜0 : Z → π −1 (B) be its lifting at t = 0.

Notice that Mn [X/A]−Mn−1 [X/A] can be thought of as the subspace of Mn [X] consisting of configurations of n distinct points, all outside A and with non-trivial labels. Therefore, we can think of B as of a subspace of Mn [X].

Define g : Z → M [X] as the difference g(z) = f˜0 (z) − f0 (z).

The map g is well-defined, continuous and its image belongs to M [A].

Since Z is compact, the image of f˜0 belongs to M [i] [X] for some i; it follows that the coefficients of g(z) are composable with the coefficients of f0 (z) in M [i] for all z ∈ Z. Lemma 3.1 guarantees the existence of a deformation dit of M [i] inside M [i+1] such that each point in di1 (M [i] ) is composable in M [i+1] with each point in the image of Z × [0, 1] under ft . There is an induced deformation of M [i] [A] which we also denote by dit .

Define the homotopy gt : Z → M [A] as dktε−1 ◦ g for 0 ≤ t < ε and as dk1 ◦ g for ε ≤ t ≤ 1. Lemma 3.1 implies that gt is well-defined and is composable with ft for all t. Consider the map f˜t = ft +gt : Z → M [X].

By construction, it lifts ft .

It remains to see that the map π is a quasifibration over some neighbourhood of Mn−1 [X/A].

Recall from Lemma 3.2 that there is a homomorphism of partial monoids M [i] → I which sends a map α to the maximal value of hi ◦ α. It gives rise to a map vi : Mn[i] [X] → In [X].

If X and A are cell complexes, let ∆ be the subspace of In [X] consisting of configurations which either contain a point of A, or have less than n points. It is not hard to show that In [X] is a cell complex and

13

Partial monoids and Dold-Thom functors

∆ is a subcomplex. In particular, ∆ is a strong deformation retract of its neighbourhood U ⊂ In [X]. Let dt : In [X] × [0, 1] → In [X] be the deformation that retracts U to ∆. We can assume that dt is a homeomorphism for all t < 1. Then dt can be lifted to a deformation t : Mn[i] [X] × [0, 1] → Mn[i] [X], D

[i]

that retracts the open subset vi−1 (U ) to the subspace vi−1 (∆) ⊂ Mn [X], which consists of configurations which either contain a point of A, or have less than n points.

Now, by construction, there exists a deformation Dt : Mn[i] [X/A] × [0, 1] → Mn[i] [X/A], such that

t Dt ◦ π = π ◦ D

for all t. In particular, Dt retracts the open neighbourhood π(vi−1 (U )) [i] [i] of Mn−1 [X/A] onto Mn−1 [X/A].

Let V be the union of all the neighbourhoods π(vi−1 (U )) for all i in M [X/A]. Then V is open in M [X/A] and it follows from Lemma 3.4 that the projection π −1 (V ) → V satisfies the conditions of Lemma 4.2.

5 5.1

On the spectrum M [S] The spectrum M [S] and the proof of Theorem 2.3.1.

Proposition 4.1 with X = Dn and A = ∂Dn gives the quasifibration M [S n−1 ] → M [Dn ] → M [S n ].

The space M [Dn ] is contractible, and therefore, we have weak homotopy equivalences M [S n−1 ] ΩM [S n ] and the spaces M [S i ] for i ≥ 0 form an Ω-spectrum which we denote by M [S].

More generally, given X, the cofibration X → CX → ΣX gives rise to a weak homotopy equivalence M [X] ΩM [ΣX], and given an inclusion map i : A → X, the cofibration A → Cyl(i) → X ∪i CA gives rise to an exact sequence . . . π∗ M [A] → π∗ M [X] → π∗ M [X ∪i CA] → π∗−1 M [A] → . . . .

14

Jacob Mostovoy

Here CX is the cone on X and Cyl(i) is the cylinder of the map i. Since π∗ M [X] is, clearly, a homotopy functor, this means that the groups π∗ M [X] form a reduced homology theory.

There is a natural transformation of the homology with coefficients in M [S i ] to π∗ M [X], induced by the obvious map M [S i ] ∧ X → M [S i ∧ X]

that sends mi zi , x to mi (zi , x):

lim πk+i M [S i ] ∧ X → lim πk+i M [S i ∧ X] = πk M [X]. i→∞

i→∞

If X is a sphere, the Freudenthal Theorem implies that this is an isomorphism. Hence, π∗ M [X] coincides as a functor, on connected cell complexes, with the homology with coefficients in M [S].

5.2

The weak equivalence of M [S] and M.

We shall first construct a weak homotopy equivalence between the infinite loop spaces of the spectra M and M [S] and then show that there exists an inverse to this equivalence, which is induced by a map of spectra. This will establish Theorem 2.3.1.

Let I be the interval [−1/2, 1/2]. By Proposition 4.1 the map I → S 1 which identifies the endpoints of I induces a quasifibration M [I] → M [S 1 ] with the fibre M Ω∞ M∞ .

Since M [I] is contractible, it follows that M is weakly homotopy equivalent to ΩM [S 1 ]; the weak equivalence is realized by the map ψ that sends m ∈ M to the loop (parameterized by I) whose value at the time t ∈ I is the configuration consisting of one point with coordinate −t and label m.

Let us now define the map of spectra Φ, inverse to the above weak equivalence ψ.

For n > 0 identify S n with the n-dimensional cube I n = [−1/2, 1/2]n ⊂ Rn , modulo its boundary. Fix a homeomorphism of the interior of I n with Rn , say, by sending each coordinate uk to tan πuk . Think of a point in M [S n ] as of a sum (xα , mα ) where xα ∈ I n and mα are maps from I n to M n . Assume that the maps mα (−xα ) are composable as maps from some Ri to M n+i . Then their sum is a well-defined point of M n which does not depend on i. We set Φn ( (xα , mα )) to be equal to this sum.

Partial monoids and Dold-Thom functors

15

The map Φn : M [S n ] → M n is only partially defined, but this problem can be circumvented as follows.

Let M ∗ [S n ] ⊂ M [S n ] be the subspace consisting of the configurations (xα , mα ) such that for some i the points mα (qα ) ∈ Ω i M n+i are composable (that is, have disjoint supports in Ri ) for any choice of the qα ∈ I n .

The spaces M ∗ [S n ] form a sub-spectrum M ∗ [S] of M [S]. Indeed,

the structure map of M [S] sends (xα , mα ) ∈ M [S n ] to the loop t →

((xα , t), mα ), where t ∈ I. The map Φn is well-defined on M ∗ [S n ]. If we define the map Φ0 simply as ΩΦ1 , it is clear from the construction that Φ0 ◦ ψ is the identity map on M0 . The proof will be finished as soon as we prove the following Proposition 5.2.1. The inclusion map M ∗ [S n ] → M [S n ] is a weak homotopy equivalence for all n > 0.

Proof. Define Mk∗ [S n ] as M ∗ [S n ] ∩ Mk [S n ]. It sufficient to prove that the inclusion Mk∗ [S n ] → Mk [S n ] is a weak homotopy equivalence for all k.

For k = 1 this inclusion is the identity map. Also, for k > 1 the map (1)

∗ Mk∗ [S n ]/Mk−1 [S n ] → Mk [S n ]/Mk−1 [S n ]

is a weak homotopy equivalence. Indeed, take a map f : S j → Mk [S n ]/Mk−1 [S n ] and let us show that the labels of the points in each configuration in the image of f can be pushed away from each other, thus deforming the ∗ [S n ]. image of f into Mk∗ [S n ]/Mk−1 By forgetting the labels in the configurations we get a map (S j − f −1 (∗)) → Bk (S n ) to the configuration space of k distinct particles in S n . Without loss of generality we can assume that the boundary of S j − f −1 (∗) is a collared (for instance, smooth) hypersurface in S j , so that this map gives rise to a bundle ξ of k-element sets over the compactification C of S j − f −1 (∗). In turn, this bundle of sets gives rise to a k-vector bundle η whose fibre is spanned by the elements of the corresponding fibre of ξ.

Since C is compact, η can be considered as a subbundle of some trivial bundle. What this means is that there exists an N such that given a configuration in the image of f we can assign a unit vector in some RN

16

Jacob Mostovoy

to each of its points so that the vectors assigned to all the points are mutually orthogonal. [i] [i] Now, the image of f is contained in Mk [S n ]/Mk−1 [S n ] for some i. We can treat the labels of the configurations in the image of f as elements of M [i+N ] = Ωi ΩN M i+N , and write them x ∈ Ri and y ∈ RN . For z ∈ C as functions m(x, y) with n write f (z) = (qα , mα ) where qα ∈ S and mα ∈ M [i+N ] , and define ft (z) as

(qα , mα (x, y + tvα )), ft (z) =

where vα are the orthogonal vectors associated to the points qα of the configuration f (z). Then for t sufficiently large, the image of ft will lie ∗ [S n ]. in Mk∗ [S n ]/Mk−1 ∗ [S n ] ⊂ M ∗ [S n ] are not, The subspaces Mk−1 [S n ] ⊂ Mk [S n ] and Mk−1 k strictly speaking, neighbourhood deformation retracts, but each of these subspaces has a neighbourhood such that the map of any compact into this neighbourhood can be retracted onto the subspace. This is sufficient to claim that the fact that the maps (1) are weak homotopy equivalences implies that the inclusion Mk∗ [S n ] → Mk [S n ] induce isomorphisms in homology. The fundamental groups of these two spaces are easily seen to be abelian for k > 1, just as in the case of the usual symmetric products, and, hence, this inclusion is a weak homotopy equivalence.

Acknowledgement I would like to thank Ernesto Lupercio for conversations that provided the motivation for this paper, and to Norio Iwase, Dai Tamaki and Peter Teichner for pointing out useful references. I am also grateful to Max-Planck-Institut f¨ ur Mathematik, Bonn for hospitality during the stay at which this paper was written.

Jacob Mostovoy Departamento de Matem´ aticas, Centro de Investigaci´ on y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 M´exico, D.F. jacob@math.cinvestav.mx

Partial monoids and Dold-Thom functors

17

References [1] Aguilar M.; Gitler S.; Prieto C., Algebraic Topology from a Homotopical Viewpoint, Springer-Verlag, 2002.

[2] Anderson D. W., Chain functors and homology theories., Sympos.

Algebraic Topology (1971), Lect. Notes Math. 249 1971, 1-12.

[3] Bousfield A. K.; Friedlander E. M., Homotopy theory of Γ-spaces, spectra, and bisimplicial sets., Geom. Appl. Homotopy Theory, II, Proc. Conf., Evanston (1977), Lect. Notes Math. 658 1978, 80-130.

[4] Dold A.; Thom R., Quasifaserungen und unendliche symmetrische Produkte., Ann. Math. 67 (1958), 239-281.

[5] Hatcher A., Algebraic Topology,. Cambridge University Press, Cambridge, 2002.

[6] Kuhn N., The McCord model for the tensor product of a space and a commutative ring spectrum., Categorical decomposition techniques in algebraic topology., Birkh¨auser, Basel, 2004.

[7] May J. P., Categories of spectra and infinite loops spaces., Lect.

Notes Math. 99 1969, 448-479.

[8] McCord M. C., Classifying spaces and infinite symmetric products., Trans. Am. Math. Soc. 146 (1969), 273-298.

[9] Voevodsky V., A1 -homotopy theory. Proceedings of the International Congress of Mathematicians, Vol. I Berlin (1998), Doc. Math. Extra Vol. I 1998, 579-604.

[10] Segal G., Configuration-spaces and iterated loop-spaces., Inventiones Math. 21 (1973), 213–221.

[11] Segal G., Categories and cohomology theories., Topology 13 (1974), 293–312.

[12] Segal G., K-homology theory and algebraic K-theory., K-Theory Oper. Algebr., Proc. Conf. Athens/Georgia (1975), Lect. Notes Math. 575 1977, 113-127.

[13] Schw¨anzl R.; Vogt R.M., E∞ -spaces and injective Γ-spaces, Manuscripta Math. 61 (1988), 203–214.

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Jacob Mostovoy

[14] Shimakawa K., Configuration spaces with partially summable labels and homology theories., Math. J. Okayama Univ. 43 (2001), 43-72.

Morfismos, Vol. 17, No. 1, 2013, pp. 19–39

´ Algebra C ∗ generada por operadores de Toeplitz con s´ımbolos discontinuos en el espacio de Bergman armo´nico ∗ Maribel Loaiza

Carmen Lozano

Resumen Sea una curva simple suave en el disco unitario complejo D. En este traba jo estudiamos el a ´lgebra de Calkin del a´lgebra C ∗ generada por operadores de Toeplitz que actu ´an en el espacio de Bergman armo ´nico de D, cuyos s´ımbolos son funciones continuas en D \ . El resultado principal e inesperado es que el espectro de un operador de Toeplitz cuyo s´ımbolo es una funcio´n constante a trozos depende del a ´ngulo de discontinuidad.

2010 Mathematics Subject Classification: 31A05,32A10,32A36,47L80.

Keywords and phrases: funci´ on arm´ onica, espacios de Berman, ´ algebras C ∗ , proyecciones de Bergman, anti-Bergman, operador de Toeplitz.

1

Intro duccio ´n

El ob jetivo de este traba jo es estudiar el a´lgebra C ∗ generada por operadores de Toeplitz en el espacio de Bergman armo´nico del disco unitario complejo. Existen numerosos trabajos sobre operadores de Toeplitz con s´ımbolo continuo y continuo a trozos actuando en el espacio de Bergman del disco unitario, entre ellos [15].

Con respecto a operadores de Toeplitz actuando en el espacio de Bergman armo ´nico, en [7] se demuestra que, mo´dulo los operadores ∗

Trabajo basado en la tesis de Carmen Lozano, dirigida por Maribel Loaiza, presentada como requisito para la obtencio ´n del grado de Maestr´ıa en Ciencias con especialidad en Matema ´ticas del CINVESTAV-IPN el 25 de enero de 2010.

19

20

Maribel Loaiza y Carmen Lozano

compactos, el ´ algebra C ∗ generada por los operadores de Toeplitz, con s´ımbolo continuo es isomorfa al ´ algebra de funciones continuas en la frontera de D. Un hecho muy conocido es que este mismo resultado es v´alido cuando los operadores de Toeplitz act´ uan en el espacio de Bergman.

En general, el comportamiento de los operadores de Toeplitz depende del espacio donde ´estos act´ uan. Uno de los resultados importantes que se tienen es que el ´ındice de un operador de Toeplitz actuando en el espacio de Bergman depende del n´ umero de vueltas que su s´ımbolo da al cero. Por otro lado, si estos operadores act´ uan en el espacio de Bergman arm´ onico, tienen siempre ´ındice cero como se demuestra en [7]. M´as aun, en este trabajo, demostramos que si el s´ımbolo de un operador de Toeplitz es continuo a trozos, su ´ındice es tambi´en cero.

Otra de las diferencias importantes entre los operadores de Toeplitz actuando en el espacio de Bergman con respecto a los mismos actuando en el espacio de Bergman arm´ onico es el exhibido en el Corolario 3.9.

En este corolario se muestra que el espectro de un operador de Toeplitz con s´ımbolo continuo a trozos depende del a´ngulo de discontinuidad en la frontera del disco.

En este trabajo se toma como base los art´ıculos [6] y [9]. En el primer art´ıculo se estudia el ´ algebra C ∗ generada por los operadores de multiplicaci´ on por funciones continuas a trozos, la proyecci´on de Bergman y la proyecci´ on anti-Bergman y; en el segundo, se estudia el ´algebra C ∗ generada por los operadores de multiplicaci´on por funciones continuas a trozos y la proyecci´ on arm´onica.

2

Preliminares

A lo largo de este trabajo D denotar´ a al disco unitario abierto en el plano complejo C, es decir, D = {z ∈ C : |z| < 1} y T denotar´a su frontera ∂D = {z ∈ C : |z| = 1}. Para z = x + iy, dm(z) = π1 dxdy es la medida de Lebesgue normalizada en D. El espacio de Bergman A2 (D) del disco D es el espacio de funciones anal´ıticas de L2 (D) y A 2 (D) = {f : f ∈ A2 (D)}, el espacio anti-Bergman, es el espacio de todas las funciones anti-anal´ıticas de L2 (D). Los espacios de Bergman y anti-Bergman son subespacios cerrados de L2 (D) = L2 (D, dm) y por lo tanto son espacios de Hilbert.

Operadores de Toeplitz con s´ımbolos discontinuos

21

La funci´ on evaluaci´ on puntual definida en A2 (D) es una funci´on continua. Por el Teorema de Representaci´on de Riesz existe un u ´nico 2 elemento kz ∈ A (D) tal que

f (z) = f (ζ)kz (ζ)dm(ζ).

D

La funci´ on K(z, ζ) = kz (ζ) se llama el n´ ucleo de Bergman de D y tiene la propiedad reproductora:

f (ζ)K(z, ζ)dm(ζ), f (z) = D

ucleo reproductor de Bergman es una funci´on para toda f ∈ A2 (D). El n´ sim´etrica hermitiana (vea por ejemplo [16]) y su f´ormula es K(z, ζ) =

1 .

(1 − zζ)2

La proyecci´ on ortogonal de L2 (D) sobre A2 (D), denotada aqu´ı por on de Bergman y est´a dada por la f´ormula BD , se llama la proyecci´ integral

f (ζ) dm(ζ).

(BD f )(z) = 2 D (1 − zζ) Por otro lado, la funci´ on n´ ucleo anti-Bergman para el caso del disco D est´a dada por la expresi´ on ζ) = K(z,

1 .

(1 − zζ)2

La proyecci´ on ortogonal de L2 (D) sobre A 2 (D) se representa en forma integral por D f )(z) = (B

D

f (ζ) dm(ζ).

(1 − zζ)2

Una funci´ on u : D → C se llama arm´onica si sus segundas derivadas parciales existen, son continuas y su Laplaciano es cero; esto es, ∆u =

∂2u ∂2u + 2 = 0, z = x + iy. ∂x2 ∂y

22

Maribel Loaiza y Carmen Lozano

El espacio de Bergman arm´ onico b2 (D) es el conjunto de todas las funciones arm´ onicas complejas u en D para las cuales u 2 =

2

D

|u(ζ)| dm(ζ)

1/2

< ∞.

El espacio b2 (D) es un subespacio cerrado de L2 (D) y por lo tanto es un espacio de Hilbert. Adem´ as cada evaluaci´on puntual es un funcional lineal acotado en b2 (D); vea por ejemplo [1]. Por lo tanto existe una u ´nica funci´ on R(z, ·) en b2 (D) la cual satisface la propiedad:

u(ζ) R(z, ζ) dm(ζ), (z ∈ D) u(z) = D

para todo u ∈ b2 (D). El n´ ucleo reproductor arm´onico R(z, ·) es real y sim´etrico. Tambi´en podemos ver que b2 (D) ∩ L∞ (D) es denso en b2 (D). Sea Q la correspondiente proyecci´on ortogonal del espacio de Hilbert L2 (D) en b2 (D). La proyecci´on de Bergman arm´onica Q tiene la representaci´ on (1)

D + T, Q = BD + B

donde T es el operador unidimensional dado por la f´ormula

(T f )(z) = − f (w) dm(w).

D

Para una funci´ on a ∈ L∞ (D) definimos el operador de Toeplitz con s´ımbolo a, Ta : b2 (D) → b2 (D) mediante la f´ormula Ta (u) = Q(au).

A continuaci´ on enunciamos algunas propiedades, cuyas demostraciones son inmediatas, que cumplen los operadores de Toeplitz en el espacio de Bergman arm´ onico.

Teorema 2.1. Sean a, b ∈ L∞ (D) y α, β ∈ C, entonces (i) Ta 2 ≤ a ∞ , (ii) Tαa+βb = αTa + βTb , (iii) Ta∗ = Ta .

Operadores de Toeplitz con s´ımbolos discontinuos

23

Actualmente existen varias t´ecnicas de localizaci´on en teor´ıa de operadores. Uno de los trabajos pioneros en esta ´area fue realizado por Simonenko ([12]) en 1965. En este trabajo se introduce la noci´on de operadores localmente equivalentes y se desarrolla lo que posteriormente se conocer´ıa como principio local de Simonenko. En el estudio de este campo surge el desarrollo de varios principios locales, entre ellos, el principio local de Douglas-Varela ([14]). En ´el se establece la representaci´ on ∗ ∗ de un ´algebra C como el espacio de secciones continuas de un haz C .

Una propiedad muy importante de los operadores de Toeplitz con s´ımbolo continuo, actuando en b2 (D), es que el conmutador y el semiconmutador de cada par de este tipo de operadores es compacto (vea por ejemplo [2], [5] y [10]). Esto nos permite utilizar el principio local de Douglas-Varela, usando como sub´ algebra central al ´algebra generada por los operadores de Toeplitz con s´ımbolo continuo. Como es usual C(D) denota al ´ algebra de las funciones continuas en D. Teorema 2.2 ([2], [10]). Sean a, b ∈ C(D), entonces el conmutador [Ta , Tb ] = Ta Tb − Tb Ta y el semiconmutador [Ta , Tb ) = Tab − Ta Tb son compactos en b2 (D).

El teorema que presentamos a continuaci´on es muy importante en el estudio del a´lgebra C ∗ generada por operadores de Toeplitz. Pues nos dice que el ideal de operadores compactos contiene a los operadores de Toeplitz cuyo s´ımbolo se anula en la frontera del disco unitario complejo.

Teorema 2.3 ([2], [10]). Si a ∈ C(D), entonces Ta es compacto en b2 (D) si y s´ olo si la restricci´ on a|T ≡ 0.

El siguiente resultado describe la relaci´on que hay entre los s´ımbolos arm´onicos de dos operadores de Toeplitz que conmutan.

Teorema 2.4 ([3]). Sean u, v ∈ b2 (D) funciones no constantes. Enolo si v = αu + β con α, β ∈ C. tonces Tu Tv = Tv Tu en b2 (D) si y s´

2.1

Las proyecciones de Bergman y anti-Bergman en el semiplano superior.

Consideremos el semiplano superior Π con la medida de ´area dz = dxdy, z = x + iy. Como es usual L2 (Π) denota al espacio de todas las funciones medibles cuadrado integrables en Π. El correspondiente espacio de Bergman de todas las funciones anal´ıticas de L2 (Π) se denotar´a por

24

Maribel Loaiza y Carmen Lozano

A2 (Π). La proyecci´ on ortogonal de L2 (Π) en A2 (Π) se denota por BΠ .

2

An´alogamente, A (Π) denota al subespacio de L2 (Π) formado por las Π a la proyecci´on ortogonal de L2 (Π) sobre funciones anti-anal´ıticas y B 2

on anti-Bergman (ver [13]). En coordenadas A (Π), llamada proyecci´ polares tenemos la descomposici´ on: (2)

L2 (R2 , dxdy) = L2 (R+ , rdr) ⊗ L2 (T, dω),

donde dω es la medida de longitud de arco en T.

D

Figura 1: La curva en el disco unitario D. Sea una curva simple suave a trozos en el disco unitario cerrado D. Sea t0 el punto donde la curva intersecta a T. Denotemos por P C(D, ) al conjunto de todas las funciones a(z), continuas en D \ que tienen l´ımite por la derecha y por la izquierda en t0 , ´estos ser´an denotados por a+ (t0 ) y a− (t0 ) respectivamente.

Sin p´erdida de generalidad podemos suponer que t0 = −1. Consideremos el operador Wφ : L2 (D) → L2 (Π), dado por la regla de correspondencia (3)

(Wφ f )(z) = f ◦ φ−1 (z) φ−1 z (z),

∂φ acil comprobar donde φ(z) = i z+1 1−z transforma a D en Π y φz = ∂z . Es f´ que Wφ es un operador unitario, autoadjunto y por tanto una isometr´ıa

Operadores de Toeplitz con s´ımbolos discontinuos

25

lineal. Consideremos tambi´en el operador unitario V : L2 (Π) → L2 (Π) definido por V = h(z)I, donde

z+i 2 (4) h(z) = . z−i Proposici´ on 2.5 ([6], [11]). BD es unitariamente equivalente a BΠ y D es unitariamente equivalente a V ∗ B Π V. B Para una funci´ on operador-valuada

L : R → B(L2 (T)), λ → L(λ),

denotaremos por I ⊗λ L(λ) al operador en B (L2 (R) ⊗ L2 (T)) dado por la f´ormula [(I ⊗λ L(λ))f ](λ, t) = [L(λ)f (λ, ·)](t), (λ, t) ∈ R × T. Π El siguiente teorema proporciona una descomposici´on de BΠ y B en t´erminos de operadores unidimensionales. Denotaremos por T+ a la intersecci´on T ∩ Π.

Π son unitariamente equivaTeorema 2.6 ([8]). Los operadores BΠ y B

lentes a las familias de operadores I⊗λ B(λ) e I⊗λ B(λ) respectivamente.

Donde para cada λ ∈ R los operadores B(λ), B(λ) ∈ B(L2 (T+ )) son las proyecciones ortogonales sobre los espacios unidimensionales generados por las funciones  2λ  tiλ−1 , λ = 0, 1−e−2πλ gλ (t) =  t√−1 , λ = 0, π

2λ t−iλ+1 , λ = 0, e2πλ −1 g λ (t) = √t , λ = 0, π

respectivamente, con t ∈ T. M´ as a´ un, gλ , g λ = 0 y B(λ)B(λ) = 0, para todo λ ∈ R.

2.2

El ´ algebra de Toeplitz

Denotaremos por K al espacio formado por todos los operadores compactos en el espacio de Bergman arm´ onico b2 (D). Sea T (C(D)) el ´alge∗ bra C generada por los operadores de Toeplitz en el espacio de Bergman

26

Maribel Loaiza y Carmen Lozano

arm´onico con s´ımbolo en C(D). El ´algebra de Toeplitz T (C(D)) es irreducible y contiene al ideal K (vea por ejemplo [7]). Adem´as cada elemento en T (C(D)) es de la forma Tv + K, donde K es un operador compacto y v ∈ b2 (D).

En [7] Kunyu Guo y Dechao Zheng probaron el siguiente resultado.

Teorema 2.7. La sucesi´ on i

j

0 −→ K −−→ T (C(D)) −−→ C(T) −→ 0 es una sucesi´ on exacta corta; esto es, el ´ algebra cociente T (C(D))/K es isom´etricamente ∗-isomorfa a C(T), donde i es la inclusi´ on y j es la on a|T . funci´ on que transforma Ta + K en la restricci´ El teorema anterior muestra similitudes entre T (C(D)) en el espacio de Bergman arm´ onico y en el espacio de Bergman, pues el mismo resultado se cumple cuando los operadores act´ uan en el espacio de Bergman A2 (D). Sin embargo el ´ındice de Fredholm de un operador de Toeplitz actuando en b2 (D) es siempre cero contrastando con el correspondiente ´ındice de Fredholm de un operador actuando en el espacio umero de vueltas que la de Bergman A2 (D), cuyo ´ındice depende del n´ funci´on s´ımbolo le da al origen.

3

El ´ algebra generada por los operadores de Toeplitz con s´ımbolo continuo a trozos

Sea TP C = T (P C(D, )) el ´ algebra C ∗ generada por los operadores de Toeplitz en el espacio de Bergman arm´onico con s´ımbolos en P C(D, ). algebra TP C es irreducible y contiene Al contener al ´ algebra T (C(D)), el ´ al ideal K. Denotaremos por π a la proyecci´on natural π : T (P C(D, )) → T P C := T (P C(D, ))/K.

Describiremos el ´ algebra de Calkin de T (P C(D, )) utilizando el Principio Local de Douglas-Varela (para detalles vea [13]).

En la representaci´ on (1) para la proyecci´on Q, T es un operador compacto de modo que, salvo una perturbaci´on compacta, la proyecci´on ˜D .

Q es la suma de las proyecciones BD y B

Operadores de Toeplitz con s´ımbolos discontinuos

27

Para a ∈ L∞ (D), denotamos por Ma : L2 (D) → L2 (D) al operador de multiplicaci´ on Ma (f ) = af.

El siguiente resultado nos permite usar al ´algebra π(T (C(D)) = algebra central conmutativa del ´algebra T (P C(D, )).

T (C(D)) como sub´

Proposici´ on 3.1. Sea a ∈ C(D) y b ∈ L∞ (D), entonces el conmutador [Ta , Tb ] = Ta Tb − Tb Ta es compacto.

D son operadores de tipo local Demostraci´ on. Los operadores BD y B

(ver [4]), por lo que Q = BD + BD + T es tambi´en de tipo local. Esto es, Q conmuta con los operadores de multiplicaci´on por funciones continuas en D m´odulo un operador compacto. Por otra parte, tenemos Ta Tb − Tb Ta = QMa QMb − QMb QMa

= QMa Mb − QMb QMa + K = QMb Ma − QMb QMa + K = QMb [I − Q]Ma + K = QMb Ha + K,

donde K ∈ K. El operador de Hankel Ha : b2 (D) → L2 (D) es compacto pues su s´ımbolo es continuo en D (ver [5]), entonces [Ta , Tb ] es compacto.

a la imagen en T P C de un operador A en el Denotaremos por A ´algebra T (P C(D, )). Por el Teorema 2.7 T (C(D))/K es isomorfo a C(T), por lo que su espacio de ideales maximales es isomorfo a T. Sea J(t) el ideal maximal en C(T) correspondiente al punto t ∈ T, es decir, J(t) = {aI + K : a ∈ C(T), a(t) = 0} .

= J(t) · T P C al ideal bilateral cerrado de T P C Denotaremos por J(t) generado por J(t). De forma an´ aloga al caso de s´ımbolos continuos, representa al ´ algebra de Calkin de T P C y πt : T P C → T P C (t) = T P C /J(t) on natural.

T P C (t) la proyecci´

algebra local de T P C en el punto Al ´algebra T P C (t) la llamamos el ´ t. Dos operadores A1 y A2 en T P C (D, ) se llamar´an localmente equivalentes en t si πt (Aˆ1 ) = πt (Aˆ2 ),

28

Maribel Loaiza y Carmen Lozano

donde Aˆ1 (respectivamente Aˆ2 ) es la imagen de A1 (A2 ) bajo π.

La descripci´ on de las ´ algebras locales T P C (t) se descompone en dos casos: los puntos t ∈ T \ {−1} y t = −1.

3.1

´ Algebras Locales del ´ algebra T P C

El resultado que mostramos a continuaci´on nos proporciona la descripci´on de las ´ algebras locales en los puntos del conjunto T \ {−1}.

Teorema 3.2. El ´ algebra local de T P C en el punto t0 ∈ T \ {−1} es isomorfa a C.

Demostraci´ on. El operador de multiplicaci´on por una funci´on a(t)I es D +K es localmente equivalente a a(t0 )I en el punto t0 . Dado que BD + B

la identidad en TP C , el operador de Toeplitz con s´ımbolo a es localmente equivalente en t0 a a(t0 )I. El isomorfismo entre T P C (t0 ) y C est´a dado por

D a(t)I → a(t0 ).

BD + B

Supondremos que la curva es tal que bajo transformaciones de M¨obius, se transforma en un rayo que sale del origen en el semiplano superior. Sin p´erdida de generalidad podemos suponer que divide al disco D en dos regiones que denotaremos por D1 y D2 .

Utilizando la Proposici´ on 2.5 y el hecho de que la funci´on h(z), definida en la f´ ormula (4), es tal que h(0) = 1 obtenemos el siguiente lema. a lgebra local T P C (t0 ) es isomorfa Lema 3.3. Sea t0 = −1. Entonces el ´ Π Wφ χD W ∗ , j = 1, 2, al ´ algebra generada por los operadores BΠ + B j φ donde Wφ est´ a dado por la ecuaci´ on (3).

Para una funci´ on a(z) ∈ P C(D) sea (5)

lim a(z) y a− (z) = z→z lim a(z). a+ (z) = z→z 0

z∈D1

0

z∈D2

Consideremos los operadores y las transformaciones definidas en (3) y (4) y sean L = φ( ), Π1 = φ(D1 ), Π2 = φ(D2 ). As´ı, Π1 = {z ∈ Π : 0 < arg z < θ},

29

Operadores de Toeplitz con s´ımbolos discontinuos

Π2 = {z ∈ Π : θ < arg z < π},

donde θ es el a´ngulo que forma la recta L con el eje X del semiplano superior Π.

D

φ

φ( )

Π

D2 t θ

Π1 = φ(D1 ) θ

D1

Figura 2: La transformaci´ on de M¨ obius φ env´ıa al disco D en el semiplano superior Π.

Lema 3.4. El ´ algebra local en un punto t = −1 es isomorfa al a ´lgebra generada por (6)

Π )χΠ I, j = 1, 2, (BΠ + B j

donde χΠj denota la funci´ on caracter´ıstica del conjunto Πj , j = 1, 2.

Demostraci´ on. Se sigue del hecho que localmente en cero V , definido en (4), es equivalente al operador identidad y del Lema 3.3.

Comenzaremos pues a describir el ´ algebra C ∗ generada por los operadores Π )χΠ I, j = 1, 2.

(BΠ + B j

De acuerdo al Teorema 2.6, esta u ´ltima ´ algebra es isomorfa al ´algebra C ∗ generada por las funciones

I , (I ⊗λ B(λ) + I ⊗λ B(λ))χ i λ∈R

actuando en L2 (R+ )⊗L2 (T+ ) y donde χi es la funci´on caracter´ıstica del arco T+ ∩ Πi , i = 1, 2. Observemos que χ1 es la funci´on caracter´ıstica del arco determinado por el ´ angulo θ, es decir, (7)

χ1 I = χθ I,

χ2 I = I − χθ I,

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Maribel Loaiza y Carmen Lozano

donde (8)

χθ (t)I =

1 0

si t ∈ [0, θ), si t ∈ [θ, π].

M´as a´ un, dado que B(λ) y B(λ) son proyecciones ortogonales entre

s´ı Q(λ) := B(λ) + B(λ) es una proyecci´on. El Teorema 2.6 nos indica

que Q(λ) es la proyecci´ on bidimensional de L2 (T+ ) sobre el subespacio a dada por H de L2 (T+ ) generado por gλ y g˜λ . Esta proyecci´on est´ (9)

gλ , Q(λ)f = f, gλ gλ + f, g λ

para f ∈ L2 (T+ ) y L2 (T+ ) = H ⊕ H ⊥ .

algebra C ∗ generada por Q(λ) y las proyecDenotemos por Uλ al ´ ciones χθ I e I − χθ I. Para analizar Uλ denotaremos por M1 y M2 a las im´agenes de las proyecciones χθ I e I − χθ I respectivamente. Observemos que Mj ∩ H = {0} y Mj ⊂ H ⊥ . Consideremos el subespacio cerrado de L2 (T+ ) (10)

H0 = (H ⊥ ∩ M1 ) ⊕ (H ⊥ ∩ M2 )

y M = H0⊥ . Descompongamos L2 (T+ ) en la suma directa (11)

L2 (T+ ) = H0 ⊕ M.

Tenemos que H ⊂ M. Consideremos las restricciones de las proyec ciones Q(λ), χθ I e I − χθ I al espacio M definidas por

|M , P1 = χθ I |M , P2 = (I − χθ I) |M .

Q = Q(λ)

Dado que χθ I e I − χθ I suman la identidad en L2 (T+ ), tenemos que P1 + P2 = I , donde I es la identidad en M. Por otra parte, como Mj ⊂ H ⊥ se sigue que todas las restricciones Pj son no triviales. Adem´as Im P1 = χθ I(M) = M1 ∩ M. De forma an´ aloga Im P2 = M2 ∩ M. Ahora, si y ∈ Mj ∩ M entonces y ∈ Pj (M) por lo que Mj ∩ M ⊂ Pj (M). Haciendo Mj = Mj ∩ M tenemos que (12)

M = M1 ⊕ M2 .

Operadores de Toeplitz con s´Ĺmbolos discontinuos

31

El conjunto {Ď&#x2021;θ IgÎť , Ď&#x2021;θ I g Îť , (I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť , (I â&#x2C6;&#x2019; Ď&#x2021;θ I) g Îť } es una base ordenada de M. Mediante el proceso de ortonormalizaci´on de GramSchmidt obtenemos la base ortonormal

f1 f2 f1 f2

e1 = , e2 = ,e = ,e = , f1 f2 1 f1 2 f2 donde f1 = Ď&#x2021;θ IgÎť , gÎť , Ď&#x2021;θ IgÎť Ď&#x2021;θ I Ď&#x2021;θ IgÎť , Ď&#x2021;θ IgÎť 2 = (I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť , gÎť , (I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť (I â&#x2C6;&#x2019; Ď&#x2021;θ I) = (I â&#x2C6;&#x2019; Ď&#x2021;θ I) gÎť â&#x2C6;&#x2019; (I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť .

(I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť 2

f2 = Ď&#x2021;θ I gÎť â&#x2C6;&#x2019; f1 f2

Tenemos adem´ as f1 2 = Ď&#x2021;θ IgÎť 2 ,

f1 2 = (I â&#x2C6;&#x2019; Ď&#x2021;θ I)gÎť 2 ,

gÎť , e1 |2 , f2 2 = Ď&#x2021;θ I g Îť 2 â&#x2C6;&#x2019; |

gÎť 2 â&#x2C6;&#x2019; | gÎť , e 1 |2 . f2 2 = (I â&#x2C6;&#x2019; Ď&#x2021;θ I)

Observemos que {e1 , e2 } es base de M1 y {e 1 , e 2 } es base de M2 .

Proposici´ on 3.5. La matriz de Q con respecto {e1 , e2 , e 1 , e 2 } de M es  ι1 ι ¯ 1 + β2 β¯1 ¯ 1 + β1 β¯1 γ1 β¯1 ι2 ι  γ¯1 β1 γ1 γ¯1 β2 γ¯1   ι1 ι ¯ 2 + β1 β¯2 γ1 β¯2 ι2 ι ¯ 2 + β2 β¯2 β1 γ¯2 γ1 γ¯2 β2 γ¯2

a la base ortonormal γ2 β¯1 γ2 γ¯1 γ2 β¯2 γ2 γ¯2



 , 

donde ι1 = e1 , gΝ , β1 = e1 , g Ν , γ1 = e2 , g Ν , ι2 = e 1 , gΝ , β2 = e 1 , g Ν y γ2 = e 2 , g Ν .

Demostraci´ on. De la ecuaci´ on (9) obtenemos las expresiones: gΝ , Q (e1 ) = e1 , gΝ gΝ + e1 , g Ν

gÎť , Q (e2 ) = e2 , g Îť

gÎť , Q (e 1 ) = e 1 , gÎť gÎť + e 1 , g Îť

gÎť .

Q (e 2 ) = e 2 , g Îť

32

Maribel Loaiza y Carmen Lozano

Dado que {e1 , e2 } es una base ortonormal de M1 , para cada y â&#x2C6;&#x2C6; M1 = on: Im Ď&#x2021;θ I se tiene la representaci´ y = y, e1 e1 + y, e2 e2 .

De igual forma cada y â&#x2C6;&#x2C6; M2 tiene una representaci´on en t´erminos de e 1 y e 2 . De esto obtenemos: Ď&#x2021;θ I(gÎť ) = gÎť , e1 e1 + gÎť , e2 e2 ,

(I â&#x2C6;&#x2019; Ď&#x2021;θ I)(gÎť ) = gÎť , e 1 e 1 + gÎť , e 2 e 2 , gÎť ) = gÎť , e1 e1 + gÎť , e2 e2 , Ď&#x2021;θ I(

gÎť ) = gÎť , e 1 e 1 + gÎť , e 2 e 2 .

(I â&#x2C6;&#x2019; Ď&#x2021;θ I)( As´Ĺ tenemos, Ď&#x2021;θ I(Q (e1 )) =

| gÎť , e1 |2 + | gÎť , e1 |2 e1

+ ( e1 , gÎť gÎť , e2 + e1 , g Îť gÎť , e2 ) e2 ,

gÎť , e1 e1 + | gÎť , e2 |2 e2 , (I â&#x2C6;&#x2019; Ď&#x2021;θ I)(Q (e2 )) = e2 , g Îť

g Îť , e1 e 1 Ď&#x2021;θ I(Q (e 1 )) = e 1 , gÎť gÎť , e1 + e 1 , g Îť

+ e 1 , gÎť gÎť , e2 + e 1 , g Îť g Îť , e2 e 2 , gÎť , e1 e1 + e 2 , g Îť gÎť , e2 e2 .

(I â&#x2C6;&#x2019; Ď&#x2021;θ I)(Q (e 2 )) = e 2 , g Îť

Representando a L2 (T+ ) como la suma directa

L2 (T+ ) = (H â&#x160;Ľ â&#x2C6;Š M1 ) â&#x160;&#x2022; (H â&#x160;Ľ â&#x2C6;Š M2 ) â&#x160;&#x2022; (M1 â&#x160;&#x2022; M2 ), tenemos que el ´ algebra UÎť es isomorfa a una sub´algebra de M4 (C). El isomorfismo est´ a dado por la transformaci´on de los generadores:

I2 0 Ď&#x2021;θ I â&#x2020;&#x2019; , 0 0

C(I â&#x2C6;&#x2019; C) C

, Q(Îť) â&#x2020;&#x2019; C(I â&#x2C6;&#x2019; C) I â&#x2C6;&#x2019;C

donde la matriz C = Cθ (Îť) est´ a dada por Cθ (Îť) = csch(ÎťĎ&#x20AC;)

eÎť(Ď&#x20AC;â&#x2C6;&#x2019;θ) senh(Νθ) Îťeâ&#x2C6;&#x2019;iθ sen θ Îťeiθ sen θ eÎť(θâ&#x2C6;&#x2019;Ď&#x20AC;) senh(Νθ)

.

Operadores de Toeplitz con s´Ĺmbolos discontinuos

33

El operador C es positivo y C â&#x2030;¤ I. Usando el Teorema Espectral obtenemos la representaci´ on para las proyecciones:

1 0 Ď&#x2021;θ I â&#x2020;&#x2019; 0 0

1 â&#x2C6;&#x2019; x x(1 â&#x2C6;&#x2019; x)

Q(Îť) â&#x2020;&#x2019; , x(1 â&#x2C6;&#x2019; x) x

con x â&#x2C6;&#x2C6; Îťâ&#x2C6;&#x2C6;R Sp(Cθ (Îť)) = Sp(Q â&#x2C6;&#x2019; Ď&#x2021;D1 )2 . Ahora el problema se reduce a hallar este espectro mediante la relaci´on:

Îťâ&#x2C6;&#x2C6;R

Sp(Cθ (Îť)) = {Sp(Cθ (Îť)) : Îť â&#x2C6;&#x2C6; R}.

Los valores propios de la matriz Cθ (Ν) est´an dados por:

x1 (Îť) = csch(ÎťĎ&#x20AC;) senh(Νθ) cosh Îť(Ď&#x20AC; â&#x2C6;&#x2019; θ) + senh2 (Νθ) senh2 Îť(Ď&#x20AC; â&#x2C6;&#x2019; θ) + Îť2 sen2 θ ,

x2 (Îť) = csch(ÎťĎ&#x20AC;) senh(Νθ) cosh Îť(Ď&#x20AC; â&#x2C6;&#x2019; θ) â&#x2C6;&#x2019; senh2 (Νθ) senh2 Îť(Ď&#x20AC; â&#x2C6;&#x2019; θ) + Îť2 sen2 θ ,

implicando que

{Sp(Cθ (Îť)) : Îť â&#x2C6;&#x2C6; R} = Im(x1 ) â&#x2C6;Ş Im(x2 ).

1 x1 (Îť)

0

x2 (Îť)

Figura 3: Gr´ afica del espectro para θ = Ď&#x20AC;/2

34

Maribel Loaiza y Carmen Lozano

1 x1 (Îť)

0

x2 (Îť)

Figura 4: Gr´ afica del espectro para θ = Ď&#x20AC;/3

Observemos que x1 (Îť) = x1 (â&#x2C6;&#x2019;Îť), es decir, es una funci´on par. M´as a´ un, θ â&#x2C6;&#x2019; sen θ , x1 (0) = Ď&#x20AC; por lo que Îť = 0 es un m´Ĺnimo local para la funci´on x1 (Îť) y un m´aximo local para la funci´ on x2 (Îť). En vista de la simetr´Ĺa de las funciones y las condiciones descritas anteriormente reduciremos nuestro an´alisis a la funci´on x1 (Îť) en el intervalo (0, â&#x2C6;&#x17E;). La derivada x 1 (Îť) de la funci´on x1 es positiva para Îť â&#x2C6;&#x2C6; (0, â&#x2C6;&#x17E;) y negativa en (â&#x2C6;&#x2019;â&#x2C6;&#x17E;, 0) por lo que 0 es un m´Ĺnimo absoluto para la funci´ on x1 (Îť). Adem´as limÎťâ&#x2020;&#x2019;Âąâ&#x2C6;&#x17E; x1 (Îť) = 1, θ , 1).

De forma an´aloga se tiene Im(x2 ) = por lo que Im(x1 ) = ( θ+sen Ď&#x20AC; θâ&#x2C6;&#x2019;sen θ (0, Ď&#x20AC; ).

Concluimos que consecuencia

Îťâ&#x2C6;&#x2C6;R

Îťâ&#x2C6;&#x2C6;R Sp(Cθ (Îť))

=

θ+sen θ θ 0, θâ&#x2C6;&#x2019;sen , 1 . En â&#x2C6;Ş Ď&#x20AC; Ď&#x20AC;

θ + sen θ θ â&#x2C6;&#x2019; sen θ â&#x2C6;Ş ,1 .

Sp(Cθ (Îť)) = 0, Ď&#x20AC; Ď&#x20AC;

As´Ĺ tenemos el resultado:

y Ď&#x2021;θ I es isomorfa a una Lema 3.6. El ´ algebra C â&#x2C6;&#x2014; generada por Q(Îť) a dado por la transforsub´ algebra de Cb (R, M2 (C)). El isomorfismo est´

Operadores de Toeplitz con s´Ĺmbolos discontinuos

35

maci´ on de los generadores:

1 0

Q(Îť) â&#x2020;&#x2019; , 0 0

1â&#x2C6;&#x2019;x x(1 â&#x2C6;&#x2019; x)

Ď&#x2021;θ I â&#x2020;&#x2019; , x(1 â&#x2C6;&#x2019; x) x θ θ ] â&#x2C6;Ş [ θ+sen , 1]. con x â&#x2C6;&#x2C6; [0, θâ&#x2C6;&#x2019;sen Ď&#x20AC; Ď&#x20AC;

Teorema 3.7. El a ´lgebra local de T P C en el punto t0 = â&#x2C6;&#x2019;1 es isomorfa al ´ algebra

C Sp(Q â&#x2C6;&#x2019; Ď&#x2021;D1 )2 .

El isomorfismo est´ a dado por la transformaci´ on de los generadores: Qa(z)IQ

â&#x2020;&#x2019;

a+ (t0 )(1 â&#x2C6;&#x2019; x) + aâ&#x2C6;&#x2019; (t0 )x,

θ θ con x â&#x2C6;&#x2C6; â&#x2C6;&#x2020; := [0, θâ&#x2C6;&#x2019;sen ] â&#x2C6;Ş [ θ+sen , 1]. Ď&#x20AC; Ď&#x20AC;

3.2

Descripci´ on del ´ algebra T P C

la compactificaci´ Denotemos por T on de T \ {â&#x2C6;&#x2019;1}. Bajo esto, al punto t+ y tâ&#x2C6;&#x2019; , ordenados en t = â&#x2C6;&#x2019;1 le corresponde un par de puntos en T, 1 1 coincide con el conjunto de ideales direcci´on positiva. El conjunto T maximales del ´ algebra T P C .

Para pegar las diferentes ´ algebras locales consideraremos los diferentes puntos en la frontera de D. El Teorema 3.7 muestra que T P C (t) θ θ ] â&#x2C6;Ş [ θ+sen , 1]), para t = â&#x2C6;&#x2019;1. es isomorfa al ´ algebra C([0, θâ&#x2C6;&#x2019;sen Ď&#x20AC; Ď&#x20AC; Para un operador compacto K, consideramos el generador del ´alge D )a(z)I + K. Usando el Teorema 3.7 podemos bra TP C : A = (BD + B definir A(x) = a+ (t)(1 â&#x2C6;&#x2019; x) + aâ&#x2C6;&#x2019; (t)x θ θ ] â&#x2C6;Ş [ θ+sen , 1]. Entonces tenemos donde x â&#x2C6;&#x2C6; [0, θâ&#x2C6;&#x2019;sen Ď&#x20AC; Ď&#x20AC;

A(0) = a+ (t), A(1) = aâ&#x2C6;&#x2019; (t).

Sea Ď&#x2018; la funci´ on que identifica los puntos de â&#x2C6;&#x2020; con los puntos de T, dada por la f´ ormula Ď&#x2018;(0) = t+ 1,

Ď&#x2018;(1) = tâ&#x2C6;&#x2019; 1.

36

Maribel Loaiza y Carmen Lozano

Figura 5: Curva Γ = ∆ ∪ϑ T y σ2 ∈ C(∆). Denotemos por S al ´algebra de todos Sean σ1 ∈ C(T) los pares (σ1 , σ2 ) que satisfacen las condiciones: (13) (14)

lim σ2 (x) = σ1 (ϑ(0))

x→t− 1

lim σ2 (x) = σ0 (ϑ(1))

x→t+ 1

La norma en el ´ algebra S est´ a determinada como sigue

σ = max sup |σ1 (t)|, sup σ2 (x) . ∆

T

Notemos que para cada par de puntos σ = (σ1 , σ2 ) define una funci´on

continua en Γ = ∆ ∪ϑ T. 0

1

t+ 1

t− 1

Figura 6: Pegado en el punto t = −1.

La descripci´ on de las ´ algebras locales T P C (t) junto con el principio local de Douglas-Varela dan lugar al resultado principal de este trabajo.

Operadores de Toeplitz con s´ımbolos discontinuos

37

algebra C ∗ T (P C(D, )) es Teorema 3.8. El ´ algebra de Calkin T P C del ´ a isomorfa e isom´etrica a C(Γ). El isomorfismo sym : TP C → C(Γ) est´ dado por la transformaci´ on de los generadores del ´ algebra TP C :

D )a(z)I + K → sym : A = (BD + B

a(t), t ∈ T, + − a (t)x + a (t)(1 − x), x ∈ ∆, t = −1,

donde a± (t) son los l´ımites definidos por la f´ ormula (5).

Finalmente, consideremos el conjunto de operadores de Fredholm de TP C y que denotaremos por Fred(TP C ).

Corolario 3.9. El ´ındice de cada operador en Fred(TP C ) es cero.

Mostraremos esto mediante el diagrama: Fred(TP C )

π Ind

G(T P C )

Z

ind

G(T P C )/G0 (T P C )

Ψ

Por un lado, G(TP C ) = G(C(Γ)) y ´este consiste de las funciones continuas en Γ que no se anulan en Γ. Calcularemos ahora la componente conexa de la identidad en G(C(Γ)), es decir, G0 (C(Γ)). Para esto observemos que cada funci´ on a ∈ C(Γ) es homot´opica a la identidad.

Entonces G0 (Γ) = G(C(Γ)) y en consecuencia G(T P C )/G0 (T P C ) = {0}.

Por lo tanto el ´ındice de Fredholm de cada operador es cero.

Maribel Loaiza Departamento de Matem´ aticas, CINVESTAV-IPN, Apartado Postal 14-740, 07000, Mexico, D.F. maribel.loaiza@gmail.com

Carmen Lozano Departamento de Matem´ aticas, CINVESTAV-IPN, Apartado Postal 14-740, 07000, Mexico, D.F. carmenlozano@math.cinvestav.edu.mx

Referencias [1] Axler S.; Bourdon P.; Ramey W., Harmonic Function Theory, Graduate Text in Mathematics 137, Springer, New York, 1992.

38

Maribel Loaiza y Carmen Lozano

[2] Choe B. R.; Lee Y. J.; Na K., Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J. 174 (2004), 165–186.

[3] Choe B. R.; Lee Y. J., Commuting Toeplitz operators on the harmonic Bergman spaces, Michigan Math. J. 46 (1999), 163–174.

[4] Dzhuraev A., Methods of Singular Integral Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics 60, Longman Scientific and Technical, 1992.

[5] Jovovi´c M., Compact Hankel operators on harmonic Bergman spaces, Integral Equations Operator Theory 22 (1995), 295–304.

[6] Karlovich Y. I.; Pessoa L., Algebras generated by the Bergman and anti-Bergman projections and by multiplications by piecewise continuous functions, Integral Equations Operator Theory, 52:2 (2005), 219–270.

[7] Guo K.; Zheng D., Toeplitz algebra and Hankel algebra on the harmonic Bergman space, J. Math. Anal. Appl. 276 (2002), 213– 230.

[8] Loaiza M., Algebras generated by the Bergman projection and operators of multiplication by piecewise continuous functions, Integral Equations Operator Theory 46 (2003), 215–234.

[9] Loaiza M., On the algebra generated by the harmonic Bergman projection and operators of multiplication by piecewise continuous functions, Bol. Soc. Mat. Mexicana 10:2 (2004), 179–193.

[10] Miao J., Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27:4 (1997), 426–438.

[11] Ram´ırez de Arellano E.; Vasilevski N., Bargmann projection, three-valued functions and corresponding Toeplitz operators, Contemp. Math. 212 (1998), 185–196.

[12] Simonenko I. B., A new general method to study linear operator equations of the singular integral equation type, II, Izv. Akad.

Nauk SSSR, Ser. Math., 29 (1965), 757–782.

[13] Vasilevski N. L., On Toeplitz operators with piecewise continuous symbols on the Bergman space, Operator Theory: Advances and Applications 170 (2007), 229–248.

Operadores de Toeplitz con s´ımbolos discontinuos

39

[14] Vasilevski N. L., C*-bundle approach to a local principle, Reporte Interno 363, Departamento de Matem´aticas, CINVESTAV del I.P.N., Mexico, 2005.

[15] Vasilevski N. L., Commutative Algebras of Toeplitz Operators on the Bergman Space, Birkh¨ auser Basel, 2008.

[16] Zhu K., Operator Theory in Function Spaces, Marcel Dekker Inc., New York and Basel, 1990.

Morfismos, Vol. 17, No. 1, 2013, pp. 41–65

Ideals, varieties, stability, colorings and combinatorial designs ∗ Javier Mun ˜ oz

1

Feliu ´ Sagols

2

Charles J. Colbourn

3

Abstract A combinatorial design is equivalent to a stable set in a suitably chosen Johnson graph, whose vertices correspond to all k-sets that could be blocks of the design. In order to find maximum stable sets of a graph G, two ideals are associated with G, one constructed from the Motzkin-Strauss formula and one reported by Lova´sz in connection with the stability polytope. These ideals are shown to coincide and form the stability ideal of G. Graph stability ideals belong to a class of 0-1 ideals. These ideals are shown to be radical, and therefore have a strong structure.

Stability ideals of Johnson graphs provide an algebraic characterization that can be used to generate Steiner triple systems.

Two different ideals for the generation of Steiner triple systems, and a third for Kirkman triple systems, are developed. The last of these combines stability and colorings.

2010 Mathematics Subject Classification: 05B07,13P10.

Keywords and phrases: computational algebraic geometry, Gr¨ obner basis, combinatorial designs, Steiner triple systems, binary ideals.

1

Introduction

Our main objective is to establish links between design theory and algebraic geometry through the use of ideals and Gro¨bner bases. We ∗

The authors thank ABACUS-CINVESTAV, CONACyT grant EDOMEX-2011C01-165873.

1 The content of this paper is part of the Ph.D. thesis of the first author working under the supervision of Feliu ´ Sagols at the Department of Mathematics of CINVESTAV. Supported by CONACyT and CINVESTAV. 2 Partially supported by SNI under contract number 7008 and CINVESTAV. 3 Partially supported by DOD grant N00014-08-1-1069.

41

42

J. MuË&#x153; noz, F. Sagols and C. J. Colbourn

concentrate on Steiner triple systems because they are simple designs with well known properties; however, the algebraic geometry techniques that we use can be easily translated to other designs.

Let us start defining the fundamental objects and concepts from design theory, graph theory and algebraic geometry with which we work.

A maximum packing by triples (MPT or MPT(n)) of order n > 0 is a maximum cardinality set of triples in {0, . . . , n â&#x2C6;&#x2019; 1} such that every pair i, j â&#x2C6;&#x2C6; {0, . . . , n â&#x2C6;&#x2019; 1} is in at most one triple. MPTs exist for every n â&#x2030;Ľ 3.

When n â&#x2030;Ą 1, 3 (mod 6), an MPT(n) is a Steiner triple system (STS or STS(n)); in this case, every 2-subset of elements appears in exactly one triple.

All graphs considered here are simple. Let v, , and i be fixed positive integers with v â&#x2030;Ľ â&#x2030;Ľ i. Let â&#x201E;Ś be a cardinality v set. Define a graph J(v, , i) as follows. The vertices of J(v, , i) are the -subsets of â&#x201E;Ś, two -subsets being v adjacent if their intersection has cardinality i.

Therefore, J(v, , i) has vertices and it is a regular graph with valency vâ&#x2C6;&#x2019; i â&#x2C6;&#x2019;i . For v â&#x2030;Ľ 2 , graphs J(v, , â&#x2C6;&#x2019; 1) are Johnson graphs [11].

One of the main methods that we use to characterize MPT(n)s consists of finding stable sets (or independent sets) in J(n, 3, 2). A stable set S of a graph G is a subset of vertices in V (G) containing no pair of adjacent vertices in G. The maximum size of a stable set in G is the stability number of G, denoted by Îą(G).

The stability polytope of a n-vertex graph G is the convex hull of {(x0 , . . . , xnâ&#x2C6;&#x2019;1 ) | xi = 1 or xi = 0 and {i â&#x2C6;&#x2C6; V (G) | xi = 1} is a stable set of G}.

We also use vertex colorings. A Îť vertex coloring (or coloring for short) of a graph G (where Îť is a positive integer) is a function c : V (G) â&#x2020;&#x2019; {1, . . . , Îť} such that (v, w) â&#x2C6;&#x2C6; E(G) if and only if c(v) = c(w).

The minimum value of Îť for which a Îť coloring of G exists is the chromatic number of G, denoted by Ď&#x2021;(G).

We introduce some algebraic structures. For k a field, k[x] = k[x1 , . . . , xn ] is the polynomial ring in n variables. A subset I â&#x160;&#x201A; k[x1 , . . . , xn ] is an ideal of k[x1 , . . . , xn ] if it satisfies 0 â&#x2C6;&#x2C6; I; if f, g â&#x2C6;&#x2C6; I, then f + g â&#x2C6;&#x2C6; I; and if f â&#x2C6;&#x2C6; I and h â&#x2C6;&#x2C6; k[x1 , . . . , xn ] then hf â&#x2C6;&#x2C6; I. When f1 , . . . , fs are polynomials in k[x1 , . . . , xn ] we set s

hi fi h1 , . . . , hs â&#x2C6;&#x2C6; k[x1 , . . . , xn ] . f1 , . . . , fs = i=1

Then f1 , . . . , fs is an ideal (see [7]) of k[x1 , . . . , xn ], the ideal gener-

Ideals and combinatorial designs

43

ated by f1 , . . . , fs . One remarkable result, the Hilbert Basis Theorem [7], establishes that every ideal I â&#x160;&#x201A; k[x1 , . . . , xn ] has a finite generating set.

The monomials in k[x] are denoted by xa = xa11 xa22 ¡ ¡ ¡ xann ; they are identified with lattice points a = (a1 , . . . , an ) in Nn , where N is the set of nonnegative integers. A total order â&#x2030;ş on Nn is a term order if the zero vector is the unique minimal element, and a â&#x2030;ş b implies a + c â&#x2030;ş b + c for all a, b, c â&#x2C6;&#x2C6; Nn .

Given a term order â&#x2030;ş, every nonzero polynomial f â&#x2C6;&#x2C6; k[x] has a unique initial monomial, denoted by inâ&#x2030;ş (f ). If I is an ideal in k[x], then its initial ideal is the monomial ideal inâ&#x2030;ş (I) := inâ&#x2030;ş (f ) : f â&#x2C6;&#x2C6; I .

The monomials that do not lie in inâ&#x2030;ş (I) are standard monomials.

A finite subset G â&#x160;&#x201A; I is a Gr¨ obner basis for I with respect to â&#x2030;ş if inâ&#x2030;ş (I) is generated by {inâ&#x2030;ş (g) : g â&#x2C6;&#x2C6; G}. If no monomial in this set is redundant, the Gr¨ obner basis is unique for I and â&#x2030;ş, provided that the coefficient of inâ&#x2030;ş (g) in g is 1 for each g â&#x2C6;&#x2C6; G. A finite subset U â&#x160;&#x201A; I is a universal Gr¨ obner basis if U is a Gr¨obner basis of I with respect to all term orders â&#x2030;ş simultaneously.

A field k is algebraically closed if for every polynomial f â&#x2C6;&#x2C6; k[x] in one variable, the equation f (x) = 0 has a solution in k. Every field k is contained in a field kÂŻ that is algebraically closed and such that every element of kÂŻ is the root of a nonzero polynomial in one variable with coefficients in k. This field is unique up to isomorphism, and is the algebraic closure of k.

Given a subset S â&#x160;&#x2020; k[x1 , . . . , xn ], the variety VkÂŻ (S) in kÂŻn is VkÂŻ (S) = {(a1 , . . . , an ) â&#x2C6;&#x2C6; kÂŻn | f (a1 , . . . , an ) = 0 f or all f â&#x2C6;&#x2C6; S}.

If I = f1 , . . . , fs â&#x160;&#x2020; k[x1 , . . . , xn ] then

VkÂŻ (I) = {(a1 , . . . , an ) â&#x2C6;&#x2C6; kÂŻn | fi (a1 , . . . , an ) = 0, 1 â&#x2030;¤ i â&#x2030;¤ s} = VkÂŻ ({f1 , . . . , fs }).

One of the most remarkable results in algebraic geometry is the following.

Theorem 1.1 (Weak Hilbert Nullstellensatz [12]). Let I be an ideal contained in k[x1 , . . . , xn ]. Then VkÂŻ (I) = â&#x2C6;&#x2026; if and only if I = k[x1 , . . . , xn ].

We may use this theorem to demonstrate that some designs do not exist, by proving that they correspond to varieties of ideals whose reduced Gr¨obner basis is {1}, or equivalently that I = k[x1 , . . . , xn ] and, by the weak Hilbert Nullstellensatz, the variety is empty.

44

J. MuË&#x153; noz, F. Sagols and C. J. Colbourn

These are the fundamental objects employed, and more specific definitions are introduced as needed. With the exception of the ideals introduced in Section 7, we use the field of rational numbers. When an algebraic closed field is needed, the complex numbers are used instead.

Computations for Gr¨ obner basis ideals are done in Macaulay 2 [9].

The paper is organized as follows. In Section 2 an ideal to generate stable sets based on the Motzkin-Strauss formula [16] is first introduced. Then, a general ideal introduced by Lov´asz [15] which has been extensively used for the generation of stable sets in graphs is described.

Both ideals are examples of 0-1 ideals, a recently introduced class having combinatorial applications beyond stability [19]. These ideals are shown to be radical, and consequently the equality of the two ideals is established. Section 3 introduces basic properties of stability ideals.

In Section 4 the stability ideal of J(n, 3, 2) is determined and used to build MPTs; difficulties to solve the equations involved are explored, and potential means to generate MPTs with restrictions are examined.

In particular, a modification of the stability ideal of J(n, 3, 2) is shown to generate anti-Pasch MPTs. Section 5 introduces two new ideals to generate MPTs that use colorings instead of stable sets. Section 6 introduces an ideal to generate Kirkman triple systems that employs a mixture of techniques based on stable sets and on colorings. Section 7 explores parametric generation of MPTs. Finally, in section 8 some concluding remarks are made.

2

Stable sets and ideals

Combinatorial and algebraic aspects of the stable set problem have been extensively studied. One of the most interesting connections is given by the Motzkin-Strauss explicit formula for ι(G) [16]: Theorem 2.1. Let G = (V, E) be a graph. Then  

 

1 = max 2 (1) xi xj xi = 1, xi â&#x2030;Ľ 0 .

1â&#x2C6;&#x2019;   Îą(G) i,j â&#x2C6;&#x2C6;E /

iâ&#x2C6;&#x2C6;V (G)

The Motzkin-Strauss formula enables one to determine part of the structure of the stability polytope, and consequently to prove several results in extremal graph theory, including Tur´anâ&#x20AC;&#x2122;s Theorem. In (1), Îą(G) is determined by an optimization problem which at first sight might be solved by Lagrange multipliers. Unfortunately the objective

Ideals and combinatorial designs

45

function reaches its maximum at the feasible region boundary and out of this region it is unbounded. We can circumvent this problem by squaring each variable to get a different version of the Motzkin-Strauss formula that still yields Îą(G):

1â&#x2C6;&#x2019;

(2)

1 Îą(G)

 

 

2 2 2 = max 2 yi yj yi = 1 .   i,j â&#x2C6;&#x2C6;E /

iâ&#x2C6;&#x2C6;V (G)

Lagrange multipliers can be used for (2). Make the objective functionâ&#x20AC;&#x2122;s gradient equal to a multiplier Îť times the restriction functionâ&#x20AC;&#x2122;s gradient to obtain the system of equations:

(3)

4yi

jâ&#x2C6;&#x2C6;V (G)|i,j â&#x2C6;&#x2C6;E /

yj2 = 2Îťyi for each i â&#x2C6;&#x2C6; V (G), yi2 = 1.

iâ&#x2C6;&#x2C6;V (G)

This system has several solutions that do not maximize (2). Lov´asz [15] characterizes the set of maximum solutions for (1): Any vector x maximizes the right hand side if and only if x has a stable set as support and if xi = 0 for some i â&#x2C6;&#x2C6; V (G) then xi = 1/Îą(G). Let y be an optimal solution to (2) such that yj â&#x2030;Ľ 0 for every j â&#x2C6;&#x2C6; V (G). From (3), if yi = 0 then

4

Îą(G) â&#x2C6;&#x2019; 1

Îą(G) Îą(G)

1 Îą(G) â&#x2C6;&#x2019; 1 = 4yi = 4 yj2 Îą(G) Îą(G) jâ&#x2C6;&#x2C6;V (G)|i,j â&#x2C6;&#x2C6;E / 1 = 2Îťyi = 2Îť Îą(G)

So, a solution of (3) is a maximum of the objective function in (2) if and only if Îť = 2 Îą(G)â&#x2C6;&#x2019;1 Îą(G) . If we substitute this value in (3), substitute 2 zi = yi Îą(G), and introduce the equations zi (zi â&#x2C6;&#x2019; 1) = 0 to restrict the values of zi to 0 or 1, then we transform (3) into

zi (zi â&#x2C6;&#x2019; 1) = 0 for each i â&#x2C6;&#x2C6; V (G),

46

(4)

J. Mu˜ noz, F. Sagols and C. J. Colbourn

zi (

j∈V (G)|i,j ∈E /

zj − α(G) + 1) = 0 for each i ∈ V (G),

i∈V (G)

zi − α(G) = 0.

This yields: Proposition 2.2. The graph G has stability number at least e if and only if the following zero-dimensional system of equations

(5)

xi (

x2i − xi = 0 f or every node i ∈ V (G),

j∈V (G)|i,j ∈E /

xj − e + 1) = 0 f or each i ∈ V (G), n

i=1

xi − e = 0,

has a solution. The vector x is a solution of (5) if and only if the support of x is a stable set.

The ideal generated by the polynomials in (5) is the Motzkin-Strauss ideal of G, denoted by M S(G).

A second approach was introduced by Lov´asz [15].

Proposition 2.3 (Lov´ asz). The graph G has stability number at least e if and only if the zero-dimensional system of equations

(6)

x2i − xi = 0 f or every node i ∈ V (G), xi xj

n

i=1

= 0 f or every edge {i, j} ∈ E(G),

xi − e = 0,

has a solution. Vector x is a solution of (6) if and only if the support of x is a stable set.

Proof. If there exists some solution x to these equations, the identities x2i − xi = 0 ensure that all variables take values only in {0, 1}. The set S = {i|xi = 1} is stable because equations xi xj = 0 guarantee that the end points of any edge in E(G) cannot belong simultaneously to S. Finally, the cardinality of S is e by the last equation.

The ideal generated by the polynomials in (6) is the stability ideal of G, denoted by S(G). As Lov´ asz [15] explains, solving (6) appears to be

Ideals and combinatorial designs

47

hopeless but he uses S(G) to write alternative proofs of several known restrictions on the stability polytope.

A quick comparison of S(G) and M S(G) demonstrates that the ideals are close; actually their generators only differ in the polynomials defined in terms of E(G). However the generators of both ideals contain the polynomials x2i − xi for i ∈ V (G). This condition confers on them a strong structure that we can generalize by introducing a bigger class of ideals containing them.

Let I be an ideal in k[x1 , . . . , xn ]. Then I is a 0-1 ideal if {x21 − x1 , x22 − x2 , . . . , x2n − xn } ⊂ I. Ideals S(G) and M S(G) are 0-1 ideals.

Our objective now is to prove that 0-1 ideals are radical, with the consequence that the Motzkin-Strauss and stability ideals are the same for any graph G. For a polynomial f ∈ k[x1 , . . . , xn ] write f = pv11 pv22 · · · pvmm where the polynomials pv11 pv22 · · · pvmm are irreducible. Polynomial f ∗ = p1 p2 · · · pm is the square free part of f . Polynomial f is square free if and only if f = f ∗.

If M is an additive group, for a natural number n and an element a of M , na denotes the n-ple sum a+· · ·+a of a (the addition of a, n times).

Under the notation, we define the characteristic of a ring k, denoted chart(k) as follows. Consider the set D = {n ∈ N|na = 0 for every a ∈ k}. If D is empty, then the characteristic of k is defined to be zero, otherwise, the least number in D is defined to be the characteristic of k. The next result is due to A. Seidenberg.

Lemma 2.4. [2, pages 341-342, 8.2] Let k be a field and let I be a zero-dimensional ideal of k[x1 , . . . , xn ], and assume that for 1 ≤ i ≤ n, I contains a polynomial fi ∈ k[xi ] with gcd(fi , fi ) = 1. Then I is an intersection of finitely many maximal ideals. In particular, I is then radical.

Proposition 2.5. [1] Let I be a zero-dimensional ideal and G be the reduced Gr¨ obner basis for I with respect to the lex term order with x1 < x2 < · · · < xn . Then we can order g1 , . . . , gt such that g1 contains only the variable x1 , g2 contains only the variables x1 and x2 and lp(g2 ) is a power of x2 , g3 contains only the variables x1 , x2 and x3 and lp(g3 ) is a power of x3 , and so forth until gn .

Here lp(g) stands for the leader power of the polynomial g.

Theorem 2.6. Let k a field and I a 0-1 ideal in k[x1 , . . . , xn ] then I is a radical ideal.

48

J. Mu˜ noz, F. Sagols and C. J. Colbourn

Proof. Let G be the reduced Gr¨ obner basis √ for I. If 1 ∈ G, by Theorem 1.1 I = k[x1 , . . . , xn ], and hence I = I. Now we consider the case when I is zero-dimensional. Since for each i = 1, . . . , n, I contains the univariate polynomial fi = x2i − xi satisfying gcd(fi , fi ) = gcd(x2i −

xi , 2xi − 1) = 1 the result follows from Lemma 2.4. √ Theorem 2.7 (Strong Hilbert Nullstellensatz). I(Vk¯ (I)) = I for all ideals I of k[x1 , . . . , xn ].

As a consequence, two ideals √ J correspond to the same variety √ I and (Vk¯ (I) = Vk¯ (J)) if and only if I = J. Proposition 2.8. For G a graph, S(G) = M S(G).

Proof. By Theorem 2.6 S(G) and M S(G) are both radical. By Propositions 2.2 and 2.3 these two ideals correspond to the same variety.

Finally, by Theorem 2.7 both ideals coincide.

Note that Proposition 2.8 is valid for all field k.

This gives two names and two ways to designate the same ideal, so henceforth the terminology of stability ideal and S(G) is used. All extremal graph theory results implied from the Motzkin-Strauss formula and those about the stability polytope can be established now from S(G). This is one reason why S(G) is important. The relevance of 0-1 ideals goes beyond stability. They help to solve problems like finding hamiltonian cycles in graphs and other combinatorial problems. A detailed presentation appears in [19].

3

Stability ideal and Gr¨ obner basis

In this section we study basic properties of the stability ideal of a graph G from the point of view of its Gr¨ obner basis. In an implicit way we use S-polynomials and Buchberger’s algorithm for the calculation of reduced Gr¨ obner basis; see [1] for details. The S-polynomial of two polynomials f and g in k[x1 , . . . , xn ], denoted S(f, g), is the polynomial ≺ (g)) ≺ (g)) · f − lcm(in≺in(f≺ ),in · g. The lcm is the least S(f, g) = lcm(in≺in(f≺ ),in (g) (g) common multiple in relation to the monomial order ≺.

We separate the generators of S(G) into sets of polynomials P1 (G) and P2 (G): (7)

P1 (G) = {x2i − xi |i ∈ V (G)}

{xi xj |i, j ∈ E(G)}

Ideals and combinatorial designs

(8)

P2 (G) = {

iâ&#x2C6;&#x2C6;V (G)

49

xi â&#x2C6;&#x2019; e}

obner Proposition 3.1. Let G be a graph. Then P1 (G) is the reduced Gr¨ basis of P1 (G) with respect to any monomial order.

Proof. Buchbergerâ&#x20AC;&#x2122;s algorithm starts with P1 (G) as initial basis. For every i, j, k, â&#x2C6;&#x2C6; V (G) with i = j and k = , S(xi xj , x xk ) = 0. If i = j then S(x2i â&#x2C6;&#x2019; xi , xi xj ) = â&#x2C6;&#x2019;xi xj . If i, j and k are pairwise different S(x2i â&#x2C6;&#x2019; xi , xj xk ) = â&#x2C6;&#x2019;xi xj xk . Finally, if i = j then S(x2i â&#x2C6;&#x2019; xi , x2j â&#x2C6;&#x2019; xj ) = â&#x2C6;&#x2019;xi (x2j â&#x2C6;&#x2019; xj ). No new polynomial should be added into the basis because any possible S-polynomial is zero or reduced to zero with respect to P1 (G). We conclude that P1 (G) is a reduced Gr¨obner basis.

The monomial order is irrelevant.

Corollary 3.2. For any G the set P1 (G) is an universal Gr¨ obner basis of P1 (G) .

This fact is a direct consequence of the following result [13].

Lemma 3.3. Let F = {f1 , f2 , . . . , fk } be a set of polynomials in k[x1 , . . . , xn ] such that polynomial fi is a product of linear factors and for any permutation Ď&#x20AC; of {1, . . . , n} we have Ď&#x20AC;(fi (x1 , . . . , xn )) = fi (xĎ&#x20AC;(1) , . . . , xĎ&#x20AC;(n) ) â&#x2C6;&#x2C6; F . If F is a Gr¨ obner basis for the ideal F with respect to the lexicographic monomial order induced by x1 > x2 > ¡ ¡ ¡ > xn then F is a universal Gr¨ obner basis for the ideal F .

The set of polynomials P1 (G) is the reduced Gr¨obner basis of P1 (G) and P2 (G) is the reduced Gr¨ obner basis of P2 (G) ; actually both of them are universal, but when we try to calculate the Gr¨obner basis of the S(G) = P1 (G) P2 (G) , the number of S-polynomials calculated by Buchbergerâ&#x20AC;&#x2122;s algorithm increases exponentially. Proposition 3.4 explains this behavior.

Proposition 3.4. The Gr¨ obner basis of S(G) with respect to the term order e < x0 < x1 < ¡ ¡ ¡ < x|V |â&#x2C6;&#x2019;1 contains the polynomial e(e â&#x2C6;&#x2019; 1)(e â&#x2C6;&#x2019; 2) . . . (e â&#x2C6;&#x2019; Îą(G)).

Proof. By Proposition 2.5 there exists a polynomial g1 in the reduced Gr¨obner basis of S(G) such that g1 is the generator of S(G)â&#x2C6;Šk[e]. Since e represents the size of the stable set this variable can be assigned to one of the values 0, 1, . . . , Îą(G). Note that g1 (i) = 0 when i â&#x2C6;&#x2C6; {0, 1, . . . , Îą(G)} / {0, 1, . . . , Îą(G)}. The polynomial e(e â&#x2C6;&#x2019; 1)(e â&#x2C6;&#x2019; and g1 (i) = 0 when i â&#x2C6;&#x2C6;

50

J. Mu˜ noz, F. Sagols and C. J. Colbourn

2) · · · (e − α(G)) has minimum degree and roots 0, 1, . . . , α(G). Thus

g1 = e(e − 1)(e − 2) · · · (e − α(G)).

If we calculate a Gr¨ obner basis for S(G), in an implicit way we are calculating α(G): Look for the polynomial in the basis containing e. This polynomial has degree α(G) + 1. Because the calculation of the stability number of a graph is NP-hard, unless P = N P , we cannot expect a polynomial time method to generate the Gr¨obner basis of S(G).

However we can use this ideal to do direct deductions related to stability.

4

Stability ideal for J(n, 3, 2) and MPTs

Maximum size stable sets in J(n, 3, 2) correspond to MPT(n)s. In this section we construct the generators of S(J(n, 3, 2)) and discuss some properties of this ideal and its Gr¨ obner basis.

Let n > 3 be an integer, and let A be a 4-set contained in Ω = {0, . . . , n − 1}. Any pair of triples in A is an edge in J(n, 3, 2). In other words, the subgraph of J(n, 3, 2) induced by the triples contained in A is isomorphic to K4 . We denote this subgraph by KA .

Proposition 4.1. Let n be a positive integer. The family {E(KA )}A is a 4-set in

Ω

is a partition of E(J(n, 3, 2)).

Proof. Let e be an arbitrary edge in E(J(n, 3, 2)), e = ({w0 , w1 , w2 }, {w0 , w1 , w3 }) for some w0 , w1 , w2 and w3 which are pairwise different elements in Ω. Then e belongs to E(K{w0 ,w1 ,w2 ,w3 } ) and E(J(n, 3, 2)) ⊆ ∪A∈{4-sets in

Ω} E(KA ).

Let A be a 4-set contained in Ω and let e be an edge of KA . There are two different triples A1 and A2 contained in A such that e = (A1 , A2 ).

We have that 4 = |A1 ∪A2 | = |A1 |+|A2 |−|A1 ∩A2 | and thus |A1 ∩A2 | = 2 or equivalently e ∈ E(J(n, 3, 2)). Thus E(KA ) ⊆ E(J(n, 3, 2)).

Finally, let B1 and B2 be different 4-sets contained in Ω, then E(KB1 ) ∩ E(KB2 ) = ∅. Suppose to the contrary that there is an edge e in the intersection of both sets. Let A1 and A2 be triples in Ω such that e = (A1 , A2 ), then A1 ∪ A2 = B1 given that e ∈ E(KB1 ), but A1 ∪ A2 = B2 because e ∈ E(KB2 ), but that is a contradiction. Thus {E(KA )}A is a 4-set in

Ω

Ideals and combinatorial designs

51

is a partition of E(B(n)).

We can use this proposition to construct the generators of S(J(n, 3, 2)).

Corollary 4.2. Let n ≥ 4 be a positive integer. Then P1 (J(n, 3, 2)) = {x2A − xA | A ⊆ {0, . . . , n − 1}

and |A| = 3} {xA xB |A, B ⊆ {0, . . . , n − 1}, (9) |A| = |B| = 3 and |A ∪ B| = 4}

P2 (J(n, 3, 2)) = { xA − e}.

A⊆T riples({0,...,n−1})

The ideal generated by the polynomials in (9) is the stability Steiner ideal of order n. We have an algorithmic approach for its construction.

Algorithm 4.1. Construction of the generators of S(J(n, 3, 2)) Input: An integer n ≥ 4.

Output: The set P of polynomials generating S(J(n, 3, 2)).

Method: 1. P ← ∅ 2. f ← 0

3. for i ← 1 to n3 4. a ← combination(n, 3, i) 5. P ← P ∪ {x2{a[0],a[1],a[2]} − x{a[0],a[1],a[2]} } 6. f ← f + x{a[0],a[1],a[2]}

7. for i ← 1 to n4 8. a ← combination(n,4,i) 9. P ← P ∪ {x{a[1],a[2],a[3]} x{a[0],a[2],a[3]} } 10. P ← P ∪ {x{a[1],a[2],a[3]} x{a[0],a[1],a[3]} } 11. P ← P ∪ {x{a[1],a[2],a[3]} x{a[0],a[1],a[2]} } 12. P ← P ∪ {x{a[0],a[2],a[3]} x{a[0],a[1],a[3]} } 13. P ← P ∪ {x{a[0],a[2],a[3]} x{a[0],a[1],a[2]} } 14. P ← P ∪ {x{a[0],a[1],a[3]} x{a[0],a[1],a[2]} } 15. P ← {f − e} 16. return P Here “combination(n, k, i)” generates (in some order) the i-th k-set contained in Ω.

52

J. Mu˜ noz, F. Sagols and C. J. Colbourn

The complexity of Gr¨ obner basis computation depends strongly on the term ordering. The best one is reported to be degree-reverselexicographical [1]; for this ordering, the computation of the Gr¨obner basis of the system of polynomial equations of degree d in n variables 2 is polynomial in dn if the number of solutions is finite [4, 5]. The time 2 needed to compute an MPT(n) is therefore polynomial in 2n . Indeed this suffices to find all possible MPT(n)s. However when n is small enough we can hope to do successful calculations to prove in “an automatic way” (through the Nullstelensatz Hilbert Theorem) conjectures about MPTs satisfying specific conditions.

We implemented this method in Macaulay 2. We adopted some heuristics, described next, that make the program faster, and use less memory to allow the computation for larger values of n.

1. Substitute the variable e in the generating set of S(J(n, 3, 2)) by the constant value of α(J(n, 3, 2)) in order to simplify computation. See [4, 5].

2. Always make the polynomials homogenous. Use reverse degreereverse-lexicographical monomial order [1].

3. Restrict the MPTs to be generated. There is no lost of generality if we assume that the MPTs contain the triples {0, 1, 2}, {0, 3, 4}, {0, 5, 6}, . . . , {0, n − 2, n − 1} and {1, 3, 5} (assuming that n is odd).

Of course, we are not working with S(J(n, 3, 2)) anymore, but we omit only systems isomorphic to those found. To enforce the presence of these triples, include in the generators the polynomials x{0,1,2} − 1, x{0,3,4} − 1, . . . , x{1,3,5} − 1. Some further pruning can be done if we consider the combined presence of other triples, for example, the pair {2, 3} could belong without loss of generality only to the triple {2, 3, 6} or to the triple {2, 3, 7}. To do this, adjoin to the generator set the polynomial x{2,3,6} +x{2,3,7} −1. We can continue with this process as desired to make the process faster and reduce the number of resulting MPTs. Taking this process to the extreme yields a full enumeration of the nonisomorphic MPTs. 4. Impose further restrictions when possible. For example, to build an anti-Pasch MPT (one not containing a copy of the MPT(6)), let a be an array containing a 6-subset of {0, . . . , n − 1}. Including x{a[3],a[4],a[5]} x{a[1],a[2],a[5]} x{a[0],a[2],a[4]} x{a[0],a[1],a[3]} with the gen-

Ideals and combinatorial designs

53

erators of S(J(n, 3, 2)) prevents the Pasch {a[3], a[4], a[5]}, {a[1], a[2], a[5]}, {a[0], a[2], a[4]}, {a[0], a[1], a[3]} from appearing in the MPTs. The other 23 monomials of this form must be included for the 6-set in a. A total of n6 24 monomials must be included in order to ensure that the MPTs generated are anti-Pasch.

Despite these heuristics, computation is far too time-consuming. Being optimistic, with a supercomputer and these heuristics, we may reach values of n as big as 21. Bigger values appear to be hopeless at present.

This time consumed by this method is not very different from brute force algorithms. Why we would prefer to use the stability ideal and a program such as Macaulay 2? The answer is simple: Some conjecture is false when the number one enters the Gr¨obner basis. Macaulay 2 can in principle produce the sequence of calculations involved. The reductions and computations of S-polynomials involved is a formal deduction, while with brute force algorithms additional work is required to get a mathematical proof. On the other hand, when a conjecture is true, the Gr¨ obner basis calculation provides a full description of the associated geometric variety. Moreover, the strong structure of the ideals, if understood well, may permit direct inferences without using the Buchberger algorithm. Sturmfels [20] used a similar development on polytopes in combinatorial optimization applications. At the moment, it is speculative that such structural results can be obtained.

5

Colorings and Steiner Triple Systems

Generation of MPTs from stability ideals is natural and could be extended to other designs. Now we turn to a different approach. Stability and colorings are closely related concepts because vertices in a colour class form a stable set. In this section we use colorings to construct STSs. First, we introduce a well known ideal to find a λ coloring of a graph G provided that λ is known in advance. Then we use two variations of this ideal to construct STSs. Lemma 5.1 (Loera [14]). Let G be a graph on n vertices, and let λ be a nonnegative integer.The graph G is λ-colorable if and only if the zero-dimensional system of equations in C[x1 , . . . , xn ] (10)

xλi − 1 = 0, f or each vertex i ∈ V (G),

54

J. Mu˜ noz, F. Sagols and C. J. Colbourn

(11) xλ−1 + xλ−2 xj + · · · + xλ−1 = 0, f or each edge {i, j} ∈ E(G), i i j has a solution. Moreover, the number of solutions equals the number of distinct λ-colorings multiplied by λ!.

The coloring ideal of λ and G is the ideal Iλ (G) of C[x1 . . . , xn ] generated by the polynomials in (10) and (11).

Note that by Theorem 2.6, the coloring ideal of λ and G is radical.

By (10) every vertex can take one of λ possible colors. Let us examine (11) more thoroughly. Denote by Pλ (x, y) the polynomial xλ−1 + xλ−2 y + · · · + y λ−1 .

Lemma 5.2. Let λ be a positive integer. If r0 and r1 are roots of unity of xλ − 1 then r0 = r1 if and only if Pλ (r0 , r1 ) = 0.

Proof. We have that (12)

xλ − y λ = (x − y)Pλ (x, y).

Since r0 and r1 are roots of unity r0λ −r1λ = 1−1 = 0. If r0 = r1 then 0 = (r0 −r1 )Pλ (r0 , r1 ), since r0 −r1 = 0 we have that Pλ (r0 , r1 ) = 0. On the other hand, if r0 = r1 then there exists an integer j ∈ {0, . . . , λ − 1} 2πj 2πj such that r0 = r1 = e λ i , and so Pλ (r0 , r1 ) = λ(e λ i )λ−1 = 0. The lemma follows.

By (11) if i, j ∈ E(G) then xi should be different to xj because otherwise Pλ (xi , xj ) would be nonzero. In other words, the color assigned to xi should be different to the color assigned to xj .

Proposition 5.3. Let n ≡ 1, 3 (mod 6) be a nonnegative integer, and ( n) λ = 32 . The zero-dimensional system of equations xλ{i,j} − 1 = 0, for every pair Pλ (x{i1 ,j1 } , x{i2 ,j2 } ) · Pλ (x{i2 ,j2 } , x{i3 ,j3 } )·

(i, j) ∈ E(Kn )

Pλ (x{i3 ,j3 } , x{i1 ,j1 } ) = 0, for each set {(i1 , j1 ), (i2 , j2 ), (i3 , j3 )} not inducing a copy of K3 in Kn

has a solution if and only if {{i, j, k}|x{i,j} = x{j,k} = x{k,i} } is an STS.

Ideals and combinatorial designs

55

Proof. Suppose that the system of equations has a solution. The value of x{i,j} is the color for the edge (i, j) in Kn . We are using as many colors as there are triples in a STS(n). If the coloring is not balanced, then some color is assigned to fewer than three edges and some color is assigned to more than 3 edges. In this way there exist edges (i1 , j1 ), (i2 , j2 ), (i3 , j3 ) and (i4 , j4 ) for which x{i1 ,j1 } = x{i2 ,j2 } = x{i3 ,j3 } = x{i4 ,j4 } . Among these four edges, there are three which do not induce a copy of K3 in Kn ; we can assume that these edges are (i1 , j1 ), (i2 , j2 ) and (i3 , j3 ). By the properties of Pλ , Pλ (x{i1 , j1 } , x{i2 , j2 } )Pλ (x{i2 , j2 } , x{i3 , j3 } )Pλ (x{i3 , j3 } , x{i1 ,j1 } ) = 0 but this contradicts the existence of a solution to the system of equations. Thus three edges receiving the same color induce a copy of K3 in Kn .

In the other direction, ordering the triples of an STS(n) as {i0 , j0 , k0 }, {i1 , j1 , k1 }, . . ., {iλ−1 , jλ−1 , kλ−1 }, and for l = 0, . . . , λ − 1 we assign to x{il ,jl } , x{jl ,kl } and x{kl ,il } the l-th λ-root of unity then the system of equations is satisfied.

The ideal generated by the polynomials in the system of equations in Proposition 5.3 is the edge coloring Steiner ideal of order n.

The stability Steiner ideal of order n associates the 3-sets in {0, . . . , n − 1} to its variables; the edge coloring Steiner ideal associates the 2sets. Does some ideal to generate STSs associate the variables to 1-sets?

The answer is affirmative, but since in an STS(n) each vertex is assigned to (n − 1)/2 triples, we need (n − 1)/2 copies of each vertex. We denote by (i, j) the j-th copy of vertex i, i = 0, . . . , n−1 and j = 1, . . . (n−1)/2.

Proposition 5.4. Let n ≡ 1, 3 (mod 6) be a nonnegative integer, and ( n) λ = 32 . The zero-dimensional system of equations xλ(i,j) − 1 = 0, for each

pair (i, j) with

Pλ (x(i1 ,j1 ) , x(i2 ,j2 ) ) · Pλ (x(i2 ,j2 ) , x(i3 ,j3 ) )·

i, j = 1, . . . , (n − 1)/2

Pλ (x(i3 ,j3 ) , x(i4 ,j4 ) ) · Pλ (x(i1 ,j1 ) , x(i3 ,j3 ) )·

Pλ (x(i1 ,j1 ) , x(i4 ,j4 ) ) · Pλ (x(i2 ,j2 ) , x(i4 ,j4 ) ) = 0, for i1 , i2 , i3 , i4 ∈ {0, . . . , n − 1} distinct and

j1 , j2 , j3 , j4 ∈ {1, . . . ,

(n − 1)/2}

56

J. MuË&#x153; noz, F. Sagols and C. J. Colbourn

PÎť (x(i,j1 ) , x(i,j2 ) ) = 0, for i â&#x2C6;&#x2C6; {0, . . . , n â&#x2C6;&#x2019; 1} and j1 , j2 â&#x2C6;&#x2C6; {1, . . . ,

(n â&#x2C6;&#x2019; 1)/2}, j1 = j2

has a solution if and only if {{i, j, k}|x(i,l1 ) = x(j,l2 ) = x(k,l3 ) for some l1 , l2 , l3 â&#x2C6;&#x2C6; {0, . . . , (n â&#x2C6;&#x2019; 1)/2}} is an STS. Proof. Analogous to the proof of Proposition 5.3.

The ideal generated by the polynomials in the system of equations in Proposition 5.3 is the vertex coloring Steiner ideal of order n.

The earlier comments for the stability Steiner ideal of order n are essentially the same for the ideals in this section. As long as the number of variables decreases the complexity of the polynomials involved increases. The final effect is that, as we expect, the practical limitations of these ideals are similar.

6

Ideals and Kirkman Triple Systems

In this section we introduce an ideal based on a combination of stability and colorings for the generation of Kirkman triple systems [6].

Let s be a positive integer and let n = 6s + 3. A Kirkman triple system of order n is a Steiner triple system with parallelism, that is, one in which the set of b = (2s + 1)(3s + 1) triples is partitioned into 3s + 1 components such that each component is a subset of triples and each of the elements appears exactly once in each component.

Proposition 6.1. Let s be a positive integer and let n = 6s + 3. The zero-dimensional system of equations x2{i,j,k} â&#x2C6;&#x2019; x{i,j,k} = 0, when {i, j, k} â&#x160;&#x201A; {0, . . . , n â&#x2C6;&#x2019; 1},

x{i,j,k} x{j,k,l} = 0, when {i, j, k}, {j, k, l} â&#x160;&#x201A; {0, . . . , n â&#x2C6;&#x2019; 1} and

{i,j,k}â&#x160;&#x2020;{0,...,nâ&#x2C6;&#x2019;1}

x{i,j,k} â&#x2C6;&#x2019; (2s + 1)(3s + 1) = 0,

i = l,

3s+1 â&#x2C6;&#x2019; 1 = 0, when {i, j, k} â&#x160;&#x201A; {0, y{i,j,k}

. . . , n â&#x2C6;&#x2019; 1},

Ideals and combinatorial designs

57

x{i,j,k} x{k,l,m} P3s+1 (y{i,j,k} , y{k,l,m} ) = 0, for every unordered couple of different

3-sets {i, j, k}, and

{k, l, m} contained in {0, . . . , n − 1}.

has a solution if and only if S = {{i, j, k}|x{i,j,k} = 1} is a Kirkman triple system.

Proof. The first three equations in the system generate the stability Steiner ideal of order n, thus the set of triples S is an STS. A new variable y{i,j,k} is introduced for each vertex {i, j, k} in J(n, 3, 2). These variables are used for coloring the elements of S; by the fourth equation each triple receives one of 3s + 1 colors. When x{i,j,k} = 0 the value of y{i,j,k} is immaterial. By the fifth equation, when x{i,j,k} = 1 the color assigned to y{i,j,k} must be different from the one assigned to every other triple in S intersecting {i, j, k}.

Using the technique in the proof of Proposition 5.3, every color is associated to exactly 2s + 1 variables yi,j,k . So S is a Kirkman triple system.

The ideal generated by the polynomials in the system of equations in Proposition 5.3 is the Kirkman ideal of order n.

In Proposition 6.1 the fifth equation is equivalent to the conditional statement: if {i, j, k} and {k, , m} are in S then Put {i, j, k} and {k, , m} in different color classes.

Few elements in the ideal suffice for the construction of ideals related to design theory: stability, colorings, Pλ polynomials and the proper use of conditional polynomial constructions.

7

Parametric generation of STSs

Let V = V(f1 , . . . , fs ) ⊂ k be a variety. Let k(t1 , . . . , tm ) represent the field of rational functions, that is, quotients between two polynomials in k[t1 , . . . , tm ]. The rational parametric representation of V consists of rational functions r1 , . . . , r ∈ k(t1 , . . . , tm ) such that the points (x1 , x2 , . . . , x ) given by

58

J. Mu˜ noz, F. Sagols and C. J. Colbourn

(13)

xi = ri (t1 , . . . , tm )

i = 1, . . . ,

lie in V . When functions r1 , . . . , r are polynomials rather than rational functions this is a parametric polynomial representation. The original defining equations f1 , . . . , fs form the implicit parametric representation of V .

It is well known that not every affine variety has a rational parametric representation; however the set of points described by a rational parametric representation is always an affine variety. In this section we consider the triples in a STS(n) as points in R3 (fixing elements in some particular order for each triple), and then we try to build a parametric polynomial representation for them. When successful, it is implicitly proved that the points produced from the triples in the STS form an affine variety.

For instance, for n = 7 the following parametric polynomial equations generate an STS(7). x = t mod 7 (14)

y = 1 + t mod 7 z = 3 + t mod 7

Taking t = 0, . . . , 6 produces the STS {{0, 1, 3}, {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 0}, {5, 6, 1}, {6, 0, 2}}.

This is a parametric polynomial representation that works exactly as we want. The polynomials in (14) belong to Z/7Z[x, y, z, t]. However, we cannot generalize this directly because the quotient ring Z/nZ is a field only when n is prime. This is a technical difficulty, addressed later.

First, let us generalize the parametric representation in (14). , . . . , n be Let n ≡ 1, 3 (mod 6) be an integer and let , l1 , l2 , l3 , n1 nonnegative integers such that ni ≤ n for i = 1, . . . , and j=1 ni = n(n − 1)/6 (the number of triples in an STS(n)). A polynomial parametric Steiner representation (PPSR) of order n, and parameters , l1 , l2 , l3 , 1 2 3 , {βi }li=0 , {δi }li=0 ), such that the n1 , . . . , n is a triple ({αi }li=0 elements in each succession are pairwise different and belong to (Z+ {0}) . We denote a parametric representation like this as P(n, , l1 , l2 , l3 , {ni } i=1 , 1 2 3 , {βi }li=0 , {δi }li=0 )). A PPSR is feasible if the system of equa({αi }li=0 tions

59

Ideals and combinatorial designs

x(t) =

l1

i=0

a αi t

αi

y(t) =

l1

i=0

b βi t

βi

z(t) =

l1

c δ i t δi

i=0

in the variables aα0 , . . . , aαl1 , bβ0 , . . . bβl2 , cδ0 , . . . , cδl3 , (where t = (t1 , . . . , tl )) has a solution such that the set S = {{x(t), y(t), z(t)}|t ∈ {0, . . . , n1 − 1} × . . . × {0, . . . , n − 1}} is an STS. That ni ≤ n for i = 1, . . . , is necessary because the operations are on Z/nZ; but it imposes restrictions on the PPSRs dealt with. For example, only for n = 7 can we have a PPSR with = 1. For any other value 1 of n it is not possible to find an integer n1 satisfying n1 < n and i=1 ni = n(n − 1)/6. In other words, it is impossible to generalize (14) for n > 7 using only one parameter t.

The important fact concerning PPSRs is that their feasibility is decided by weak Hilbert Nullstelensatz Theorem.

Proposition 7.1. Let n ≡ 1, 3 (mod 6) be a prime. Let P(n, , l1 , l2 , 1 2 3 , {βi }li=0 , {δi }li=0 )) be a PPSR of order n. Let P and l3 , {ni } i=1 , ({αi }li=0 Q be the polynomials in Z/nZ[aα0 , . . . , aαl1 , bβ0 , . . . , bβl2 , cδ0 , . . . , cδl3 ], P (u) = (u − 1)(u − 2) · · · (u − n + 1), Q(u) = uP (u), u ∈ {0, . . . , n − 1}.

Then P is feasible if and only if the zero-dimensional system of equations  Q(aαi )  for i = 0, . . . , l1 , Q(bβj ) = 0, j = 0, . . . , l2 and (15)  k = 0, . . . , l3 Q(cδk )  P (x(t) − y(t))  for t ∈ {0, . . . , n1 − 1} P (x(t) − z(t)) (16) = 0, × . . . × {0, . . . , n − 1}  P (y(t) − z(t))  P (x(t1 ) − x(t2 ))P (y(t1 ) − y(t2 ))    P (x(t1 ) − y(t2 ))P (y(t1 ) − x(t2 ))    for t1 , t2 ∈ {0, . . . ,  P (x(t1 ) − x(t2 ))P (z(t1 ) − z(t2 )) = 0, n1 − 1} × . . . × {0, . . . , P (x(t1 ) − z(t2 ))P (z(t1 ) − x(t2 ))   n − 1}, t1 = t2   P (z(t1 ) − z(t2 ))P (y(t1 ) − y(t2 ))    P (z(t1 ) − y(t2 ))P (y(t1 ) − z(t2 )) (17) has a solution.

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J. MuË&#x153; noz, F. Sagols and C. J. Colbourn

Proof. Assume that the system of equations is satisfied. Then by (15) the values of these coefficients should be in the set {0, 1, . . . , n â&#x2C6;&#x2019; 1} which corresponds to the roots of the polynomial Q(t). Also (16) guarantees that the elements in each of the triples in S are distinct. (The polynomial P plays a similar role to that of the polynomials PÎť introduced in Section 5.) Finally, by (17) every pair of different vertices in {0, . . . , nâ&#x2C6;&#x2019;1} appears in exactly one of the triples and thus it is an STS. The converse is immediate.

The ideal generated by the polynomials in Proposition 7.1 is the parametric Steiner ideal of P. Solutions to the polynomials in the parametric Steiner ideal of a PPSR can be found using Gr¨ obner bases. For example, the Gr¨obner basis for the unique possible PPSR of order n = 7 and = l1 = l2 = l3 = 1 is { c61 â&#x2C6;&#x2019; 1, b1 â&#x2C6;&#x2019; c1 , a1 â&#x2C6;&#x2019; c1 , c70 â&#x2C6;&#x2019; c0 , b60 + b50 c0 + b40 c20 + b30 c30 + b20 c40 + b0 c50 + c60 â&#x2C6;&#x2019; 1, a50 + a40 b0 + a40 c0 + a30 b20 + a30 b0 c0 + a30 c20 + a20 b30 + a20 b20 c0 + a20 b0 c20 + a20 c30 + a0 b40 + a0 b30 c0 + a0 b20 c20 + a0 b0 c30 + a0 c40 + b50 + b40 c0 + b30 c20 + b20 c30 + b0 c40 + c50 }.

A solution that makes all these polynomials zero is a0 = 0, b0 = 1, c0 = 3, a1 = 1, b1 = 1, and c1 = 1; it corresponds to the PPSR in (14).

Corollary 7.2. A PPSR P is feasible if and only if the Gr¨ obner basis of the parametric Steiner ideal of P does not contain 1.

While these provide a relatively simple way to determine the feasibility of a PPSR, it is limited to prime orders. We can circumvent this limitation by working in the complex number field. We carry the operations from Z/nZ to C through the transformation Ď&#x2020; : Z/nZ â&#x2020;&#x2019; C, 2Ď&#x20AC;k Ď&#x2020;(k) = e n i . Two well known properties of Ď&#x2020; are: For every a and b in Z/nZ (18)

Ď&#x2020;(a + b) = Ď&#x2020;(a)Ď&#x2020;(b) Ď&#x2020;(a ¡ b) = Ď&#x2020;(a)b = Ď&#x2020;(b)a

Let n â&#x2030;Ą 1, 3 (mod 6) be a prime. Let P(n, , l1 , l2 , l3 , {ni } i=1 , 1 2 3 , {βi }li=0 , {δi }li=0 )) be a PPSR of order n. We extend the do({Îąi }li=0

main of Ď&#x2020; to the polynomial x(t) = lj=1 aÎąj tÎąj as Ď&#x2020;( lj=1 aÎąj tÎąj ) =

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Ideals and combinatorial designs

αj = lj=1 a ˆtαj . This extension is compatible with (18); it takes a polynomial on the variables aα0 , . . . , aαl1 and transforms ˆαl1 (here a ˆαj stands it into a polynomial on the variables a ˆ α0 , . . . , a for φ(aαj )). For each t ∈ {0, . . . , n1 − 1} × . . . × {0, . . . , n − 1}, φ(x(t)(aα0 , . . . , aαl1 )) = φ(x(t))(ˆ a α0 , . . . , a ˆαl1 ). Similar extensions are made to φ in order to be applied to the polynomials y(t) and z(t). l

j=1 φ(aαj )

tαj

Proposition 7.3. Let n ≡ 1, 3 (mod 6) be a prime. Let P(n, , l1 , l2 , 1 2 3 , {βi }li=0 , {δi }li=0 )) be a PPSR of order n. Let Pn l3 , {ni } i=1 , ({αi }li=0 and Qn be polynomials in C[ˆ a0 , . . . , a ˆl , ˆb0 , . . . , ˆbl , cˆ0 , . . . , cˆl ], Pn (u, v) = n−1 n−2 n−2 n−1 +u v +. . .+vw +w , Qn (u) = un −1, u, v ∈ {0, . . . , n−1}. u Then P is feasible if the zero-dimensional system of equations (19)

Qn (ˆ aαi ) = Qn (ˆbβj ) = Qn (ˆ c δk )

 Pn (φ(x(t)), φ(y(t)))  Pn (φ(x(t)), φ(z(t))) (20)  Pn (φ(y(t)), φ(z(t)))  Pn (φ(x(t1 )), φ(x(t2 )))Pn (φ(y(t1 )), φ(y(t2 )))    Pn (φ(x(t1 )), φ(y(t2 )))Pn (φ(y(t1 )), φ(x(t2 )))     Pn (φ(x(t1 )), φ(x(t2 )))Pn (φ(z(t1 )), φ(z(t2 ))) Pn (φ(x(t1 )), φ(z(t2 )))Pn (φ(z(t1 )), φ(x(t2 )))    Pn (φ(z(t1 )), φ(z(t2 )))Pn (φ(y(t1 )), φ(y(t2 )))     Pn (φ(z(t1 )), φ(y(t2 )))Pn (φ(y(t1 )), φ(z(t2 )))

=

for i = 0, . . . , l1 , 0, j = 0, . . . , l2 and k = 0, . . . , l3

=

0,

=

for t1 , t2 ∈ {0, . . . , 0, n1 − 1} × . . . × {0, . . . , n − 1}, t1 = t2

for t ∈ {0, . . . , n1 − 1} × . . . × {0, . . . , n − 1}

(21)

has a solution in a ˆ0 , . . . , a ˆl1 , ˆb0 , . . . , ˆbl2 , cˆ0 , . . . , cˆl3 if and only if P is feasible.

Proof. Assume that the system of equations has a solution. From (19) ˆ , ˆb0 , . . . , ˆb , cˆ0 , . . . , cˆ could only be assigned to nth roots of a ˆ0 . . . , a unity. Since φ(x(t)), φ(y(t)), and φ(z(t)) are expressed as products and integer powers of nth roots of unity, they evaluate to nth roots of unity too. The polynomial Pn is the polynomial Pλ , with λ = n, defined in Section 5, and so, by Lemma 5.2 the arguments in the proof of Proposition 7.1 with respect to (16) and (17) are applicable to (20) and (21), respectively. So Sˆ = {{φ(x(t)), φ(y(t)), φ(z(t))}|t ∈ {0, . . . , n1 − 1} × · · · × {0, . . . , n − 1}} contains only triples of nth roots of unity and each pair of nth roots of unity is contained in exactly one triple.

When we apply φ−1 to the elements in every triple in Sˆ we obtain an STS S.

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From a computational point of view, the Gr¨obner basis of the ideal in Proposition 7.1 can be found faster in Macaulay 2 than the corresponding Gr¨ obner basis for Proposition 7.3. For n = 7 and = 1 we required with the former approach 12 seconds, with the last one the system exhausted the memory.

Now we do the same type of transformation done from Proposition 7.1 to Proposition 7.3 in the opposite direction to get an ideal on Z/nZ to obtain a λ-coloring of a graph G. We transform Lemma 5.1 in the following way.

Lemma 7.4. Let G be a graph on n vertices for some prime n, and let λ be a nonnegative integer. Graph G is λ-colorable if and only if the following zero-dimensional system of equations in Z/nZ[x1 , . . . , xn ] (22)

xi (xi − 1) · · · (xi − λ) = 0, f or every vertex i ∈ V (G),

(23) (xi − xj − 1) · · · (xi − xj − λ) = 0, f or every edge {i, j} ∈ E(G),

has a solution.

This new ideal is useful only for prime values of n but the calculation of its Gr¨obner basis is more efficient.

8

Conclusions

When Hilbert submitted his famous finiteness theorem [7] to the Mathematische Annalen in 1888, Gordan rejected the article. Gordan had earlier established the finiteness of generators for binary forms using a complex computational approach. He expected not only a finiteness existence proof, but also a more constructive approach. Gordan comment about Hilbert’s work was “Das ist nicht Mathematik. Das ist Theologie” (This is not Mathematics. This is Theology) [10]. Encouraged by Gordan’s opinion, Hilbert provided estimates of the maximum degree of the minimum set of generators. But in 1899 Gordan developed a constructive proof of the finiteness theorem, using what is now called the Gr¨obner basis to reduce to the more easily treated monomial case.

Gordan’s tools were made more practical with the advent of modern computers. Despite this, implicit in the calculation of many Gr¨obner bases is the solution of NP-complete problems. Hence we cannot hope to solve every possible problem stated with Gr¨obner bases. Nevertheless, important problems in physics, robotics and engineering have been successfully solved with them.

Ideals and combinatorial designs

63

Characterizations of combinatorial designs test these algebraic tools.

We have examined how to represent the rich structure of designs into algebraic terms. We tested in Macaulay 2 that every ideal works as described. Unfortunately, the large dimensions of the systems of polynomials involved make manipulation impractical from a computational point of view. The development of parallel algorithms to calculate Gr¨obner basis efficiently are remarkable [3, 17]. Such advances may permit the direct calculation for the ideals introduced in this paper for small values of n. On the other hand, the increasing industrial interest in Gr¨obner basis will bring in the near future computer hardware especially designed to making fast the calculations involved. This progress will be important for design theory.

We opened unexplored connections between these algebraic geometry and combinatorial design theory; this is the main contribution of our work. From the algebraic geometry point of view the most interesting result from these connections is the discovery of 0-1 ideals whose structural properties and applications in combinatorics are explored in [19].

Acknowledgement We would like to express our thanks to the anonymous referee who recommended us to extend Theorem 2.6 to any arbitrary field. Originally we only have proved it for algebraically closed fields with characteristic zero.

Javier Mu˜ noz Departamento de Matem´ aticas, CINVESTAV del I.P.N, Apartado Postal 14-740, M´exico D.F,C.P.07360. jmunoz@math.cinvestav.mx

Feli´ u Sagols Departamento de Matem´ aticas, CINVESTAV del I.P.N, Apartado Postal 14-740, M´exico D.F,C.P.07360. fsagols@math.cinvestav.mx

Charlie J. Colbourn School of Computing, Informatics and Decision Systems, Arizona State University, Tempe, AZ 85287-8809, U.S.A. Charles.Colbourn@asu.edu

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References [1] Adams W.; Loustaunau P., An Introduction to Gr¨obner Bases, Graduate studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1994.

[2] Becker T.; Weispfenning V.; et al, Gr¨obner Bases. A Computational Approach to Commutative Algebra, Springer Graduate texts in Mathematics, Springer-Verlag, New York, 1993.

[3] Ajwa I. A., A case study of grid computing and computer algebra parallel Gr¨ obner bases and characteristic sets. The Journal of Supercomputing 41:1 (2007), 53–62.

[4] Caniglia L.; Galligo A.; Heintz J., Some new effectivity bounds in computational geometry, Proceedings of AAECC-6, Rome, Lecture Notes in Computer Science, 357 1988, 131–151.

[5] Caniglia L.; Galligo A.; Heintz J., Equations for the projective closure and effective Nullstellensatz, Discrete Applied Math 33 (1991), 11–23.

[6] Colbourn C. J., Triple Systems. Chapter II.2 in Handbook of Combinatorial Designs, Second Edition, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2007.

[7] Cox D.; Little J.; O’Shea D., Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.

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[9] Eisenbud D.; Grayson D. R.; Stillman M.; Sturmfels B., Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computations in Mathematics 8, Springer-Verlag, New York, 2002.

[10] Ewald W. B., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2, Oxford University Press, 1996.

[11] Godsil C. D.; Royle G., Algebraic Graph Theory, Springer-Verlag, New York, 2001.

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[12] Hungerford T. W., Algebra, Graduate Texts in Mathematics, Springer-Verlag, New York, 1974.

[13] de Loera J. A., Gr¨ obner bases and graph colorings, Beitr¨age zur Algebra und Geometrie 36 (1995), 89–86.

[14] de Loera J. A.; Lee J.; Margulies S.; Onn S., Expressing combinatorial optimization problems by systems of polynomial equations and the Nullstellensatz, axXiv:0706.0578v1 [math.CO], (5 Jun 2007).

[15] Lov´asz L., Stable sets and polynomials, Discrete Math. 124, (1994), 137–153.

[16] Motzkin T. S.; Strauss E. G., Maxima for graphs and a new proof of a theorem of Tur´ an, Canad. J. Math. 17, (1965), 533–540.

[17] Mutyunin V. A.; Pankratiev E. V., Parallel algorithms for Gr¨ obner bases construction, J. Math. Sci. 142:4, (2007), 2248– 2266.

[18] Seidenberg A., Constructions in algebra, Trans. Amer. Math. Soc. 197, (1974), 273–313.

[19] Sagols F.; Mu˜ noz J., Structural properties and applications of binary ideals. In process.

[20] Sturmfels B., Gr¨ obner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Soc., Povidence RI, 1995.

Morfismos se imprime en el taller de reproducci´ on del Departamento de Matem´ aticas del Cinvestav, localizado en Avenida Instituto Polit´ecnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, M´exico, D.F. Este n´ umero se termin´ o de imprimir en el mes de julio de 2013. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contents - Contenido Partial monoids and Dold-Thom functors Jacob Mostovoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

´ Algebra C ∗ generada por operadores de Toeplitz con s´ımbolos discontinuos en el espacio de Bergman armo ´nico Maribel Loaiza and Carmen Lozano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Ideals, varieties, stability, colorings and combinatorial designs Javier Mun ˜oz, Feliu ´ Sagols, and Charles J. Colbourn . . . . . . . . . . . . . . . . . . . . 41