Lusternik-Schnirelmann category of simplicial complexes and finite spaces

LUSTERNIK-SCHNIRELMANN CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES Abstract. In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and it only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of LScategory are obtained in the setting of finite topological spaces. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Simplicial complexes 4 2.2. Finite topological spaces 5 2.3. Associated spaces and complexes 5 2.4. LS-category 6 3. LS-category of simplicial complexes 7 3.1. Simplicial category 7 3.2. Homotopical invariance 7 4. Geometric category 9 4.1. Simplicial geometric category 9 4.2. Behaviour under strong collapses 9 5. LS-category of finite spaces 10 5.1. Maximal elements 11 5.2. Geometric category 11 Date: March 6, 2015. 2010 Mathematics Subject Classification. 55U10, 55M30, 06F30. Key words and phrases. simplicial complex, contiguity class, strong collapse, LusternikSchnirelmann category, finite topological space, poset. The first and the third authors are partially supported by PAIDI Research Groups FQM-326 and FQM-189. The second author was partially supported by MINECO Spain Research Project MTM2013-41768-P and FEDER. 1 arXiv:1501.07540v2 [math.AT] 5 Mar 2015 2 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES 6. Relation between categories 15 Acknowledgements 17 References 17 1. Introduction Lusternik-Schnirelmann category was originally introduced as a tool for variational problems on manifolds. Nowadays it has been reformulated as a numerical invariant of topological spaces and has become an important notion in homotopy theory and many other areas [5], as well as in applications like topological robotics [8]. Many papers have appeared on this topic and the original definition has been generalized in a number of different ways. For simplicial complexes and simplicial maps, the notion of contiguity is considered as the discrete version of homotopy. However, although these notions are classical ones, the corresponding theory of LS-category is missing in the literature. This paper can be considered as a first step in this direction. Still more important, finite simplicial complexes play a fundamental role in the so-called theory of poset topology, which connects combinatorics to many other branches of Mathematics [11, 15]. Being more precise, such theory allows us to establish relations between simplicial complexes and finite topological spaces. On the one hand, finite T0-spaces and finite partially ordered sets are equivalent categories (notice that any finite space is homotopically equivalent to a T0-space). On the other hand, given a finite topological space Xthere exists the associated simplicial complex K(X), where the simplices are its non-empty chains; and, conversely, given a finite simplicial complex Kthere is a finite space χ(K), the poset of simplices of K, such that K(χ(K)) = sd K, the first barycentric subdivision of K. By these constructions we can see posets and simplicial complexes as essentially equivalent objects. In this work we introduce a natural notion of LS-category scat Kfor any simplicial complex K. Unlike other topological notions established for the geometric realization of the complex, our approach is directly based on the simplicial structure. In this context, contiguity classes are the combinatorial analogues of homotopy classes. For instance, different simplicial approximations to the same continuous map are contiguous and the geometric realizations of contiguous maps are homotopic. Analogously to the topological setting, it is desirable that this notion of category be a homotopy invariant. In order to obtain this goal, the notion of strong collapse introduced by Minian and Barmak [2] is used instead of the classical notion of collapse. The existence of cores or minimal complexes is a fundamental difference between strong homotopy types and simple homotopy types. A simplicial complex can collapse to non-isomorphic subcomplexes. However if a complex Kstrongly collapses to a minimal complex K0, LS-CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES 3 it must be unique, up to isomorphism. We prove the homotopical invariance of simplicial category, and, in particular, that scat K= scat K0. In addition, a notion of geometric category gscat Kis introduced in the simplicial context. For topological spaces geometric category is not a homotopical invariant, so it is customary to consider the minimum value of gcat Y, for all spaces Yof the same homotopy type as X. This process leads to a homotopical invariant, Cat X, first introduced by Ganea [5]. In the simplicial context we prove several results about the behaviour of gscat Kunder strong collapses. Other authors [1] have considered a notion of geometric category for simple collapses. The essential difference is that for gcat one can consider not only the minimum value in the homotopy class, but also the maximum, which coincides with the category of the core of the complex. By means of the equivalence between simplicial complexes and finite topological spaces, we get a notion of LS-category of finite spaces which corresponds with the classical notion, because the concept of strong homotopy equivalence in the simplicial context corresponds to the notion of homotopy equivalence in the setting of finite spaces. Under this point of view new results are obtained which do not have analogues in the continuous case. The paper is organized as follows. We start by introducing in Section 2 the basic notions and results concerning the link between simplicial complexes and finite topological spaces, as well as the definition of classical LS-category. Section 3 is focused on the study of the simplicial LS-category scat Kof a simplicial complex K. We prove that this notion is a homotopy invariant, that is, two strongly equivalent complexes have the same category. The corresponding notion of geometrical category gcat Kfor a simplicial complex Kis studied in Section 4. We obtain that the geometrical category increases under strong collapses, and that the maximum value is obtained for the core K0of the complex. Section 5 contains a study on the LS-category of finite topological spaces. Notice that it is not the LS-category of the geometric realization |K(X)|of the associated simplicial complex, but it is the category of the topological space Xitself. We have not found any specific study of LS-category for finite topological spaces in the literature. For instance, we prove that the number of maximal elements minus one is an upper bound of the category of a finite topological space. By analogy with the LS-category of simplicial complexes we establish other results for finite spaces. For instance, we prove that geometrical category increases when a beat point is erased. In particular, we exhibit a new example showing that geometrical category is not a homotopy invariant. This example was communicated to the authors by J. Barmak and G. Minian. Finally, in Section 6 we prove that both the category and the geometrical category decrease when applying the functors Kand χ. 2. Preliminaries 4 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES 2.1. Simplicial complexes. We recall the notions of contiguity and strong collapse. Let K, L be two simplicial complexes. Two simplicial maps ϕ, ψ :K→Lare contiguous [13, p. 130] if, for any simplex σ∈K, the set ϕ(σ)∪ψ(σ) is a simplex of L; that is, if v0, . . . , vkare the vertices of σ then the vertices f(v0), . . . , f(vk), g(v0), . . . , g(vk) span a simplex of L. This relation, denoted by ϕ∼cψ, is reflexive and symmetric, but in general it is not transitive. Definition 2.1. Two simplicial maps ϕ, ψ :K→Lare in the same contiguity class, denoted by ϕ∼ψ, if there is a sequence ϕ=ϕ0∼c· · · ∼cϕn=ψ of contiguous simplicial maps ϕi:K→L, 0 ≤i≤n. A simplicial map ϕ:K→Lis a strong equivalence if there exists ψ:L→ Ksuch that ψ◦ϕ∼idKand ϕ◦ψ∼idL. We write K∼Lif there is a strong equivalence between the complexes Kand L. In the nice paper [3] Barmak and Minian showed that strong homotopy types can be described by certain types of elementary moves called strong collapses. A detailed exposition is in Barmak’s book [2]. These moves are a particular case of the well known notion of simplicial collapse [7]. Definition 2.2. A vertex vof a simplicial complex Kis dominated by another vertex v06=vif every maximal simplex that contains valso contains v0. If vis dominated by v0then the inclusion i:K\v⊂Kis a strong equivalence. Its homotopical inverse is the retraction r:K→K\vwhich is the identity on K\vand such that r(v) = v0. This retraction is called an elementary strong collapse from Kto K\v, denoted by K&& K\v. Astrong collapse is a finite sequence of elementary collapses. The inverse of a strong collapse is called a strong expansion and two complexes Kand L have the same strong homotopy type if there is a sequence of strong collapses and strong expansions that transform Kinto L. Example 2.3. Figure 1 is an example of elementary strong collapse. v0 v v0v0 Figure 1. Elementary strong collapse The following result states that the notions of strong homotopy type and strong equivalence (via contiguity) are the same. LS-CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES 5 Theorem 2.4. [3, Cor. 2.12] Two complexes Kand Lhave the same strong homotopy type if and only if K∼L. 2.2. Finite topological spaces. We are interested in homotopical properties of finite topological spaces. First, let us recall the correspondence between finite posets and finite T0-spaces. If (X, ≤) is a partially ordered finite set, we consider the T0topology on Xgiven by the basis {Ux}x∈X where Ux={y∈X:y≤x}. Conversely, if (X, τ) is a finite topological space, let Uxbe the minimal open set containing x∈X. Then we can define a preorder by saying x≤y if and only if Ux⊂Uy. This preorder is an order if and only if τis T0. Under this correspondence, a map f:X→Ybetween finite T0-spaces is continuous if and only if fis order preserving. Order spaces are also called “Alexandrov spaces”. Proposition 2.5. Any (finite) topological space has the homotopy type of a (finite) T0-space. Proof. Take the quotient by the equivalence relation: x∼yif and only if Ux=Uy. From now on, we shall deal with finite spaces which are T0. Proposition 2.6. [16] The connected components of Xare the equivalence classes of the equivalence relation generated by the order. We now consider the notion of homotopy. Let f, g :X→Ybe two continuous maps between finite spaces. We write f≤gif f(x)≤g(x) for all x∈X. Proposition 2.7. [3] Two maps f, g :X→Ybetween finite spaces are homotopic, denoted by f≃g, if and only if they are in the same class of the equivalence relation generated by the relation ≤between maps. Corollary 2.8. The basic open sets Ux⊂Xare contractible. Proof. Since the point xis a maximum of Ux, it is a deformation retract of Uxby means of the constant map r=x:Ux→Ux. 2.3. Associated spaces and complexes. To each finite poset Xthere is associated the so-called order complex K(X). It is the simplicial complex with vertex set Xand whose simplices are given by the finite non-empty chains in the order on X. Moreover, if f:X→Yis a continuous map, the associated simplicial map K(f): K(X)→ K(Y) is defined as K(f)(x) = f(x) for each vertex x∈X. Proposition 2.9. [2, Prop. 2.1.2, Th. 5.2.1] (1) If f, g :X→Yare homotopic maps then the simplicial maps K(f) and K(g)are in the same contiguity class. 6 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES (2) If two T0-spaces X, Y are homotopy equivalent, then the complexes K(X)and K(Y)have the same strong homotopy type. Notice that the reciprocal statements are not necessarily true, because two non-homotopic maps f, g may induce maps K(f),K(g) which are in the same contiguity class, through simplicial maps which do not preserve order. Conversely, it is possible to assign to any finite simplicial complex Kits Hasse diagram or face poset, that is, the poset of simplices of Kordered by inclusion. If ϕ:K→Lis a simplicial map, the associated continuous map χ(ϕ): χ(K)→χ(L) is given by χ(ϕ)(σ) = ϕ(σ), for any simplex σof K. Proposition 2.10. [2, Prop. 2.1.3, Th.5.2.1] (1) If the simplicial maps ϕ, ψ :K→Lare in the same contiguity class then the continuous maps χ(ϕ), χ(ψ)are homotopic. (2) If two finite simplicial complexes K, L have the same strong homotopy type, then the associated spaces χ(K), χ(L)are homotopy equivalent. 2.4. LS-category. We recall the basic definitions of Lusternik-Schnirelmann theory. Well known references are [5] and [10]. An open subset Uof a topological space Xis called categorical if Ucan be contracted to a point inside the ambient space X. In other words, the inclusion U⊂Xis homotopic to some constant map. Definition 2.11. The Lusternik-Schnirelmann category, cat X, of Xis the least integer n≥0 such that there is a cover of X by n+ 1 categorical open subsets. We write cat X=∞if such a cover does not exist. Category is an invariant of homotopy type. Another interesting notion, the geometric category, denoted by gcat X, can be defined in a similar way using subsets of Xwhich are contractible in themselves, instead of contractible in the ambient space X. By definition, cat X≤gcat X. However, geometric category is not a homotopy invariant [5, p. 79]. Remark 1.For ANRs one can use closed covers, instead of open covers, in the definition of LS-category. However, these two notions would lead to different theories in the setting of finite spaces. For instance, for the finite space of Example 5.2, we obtain different values for the corresponding categories. This work is limited to the nowadays most common definition of LS-category, that is, using categorical open subsets. Remark 2.Actually, the definition of LS-category by covers is not well-suited for many constructions in homotopy theory. This led to alternative definitions (Ganea, Whithead [5]) which are well known in algebraic topology. However, those constructions require that the space Xsatisfies some additional properties. One of them, the existence of non-degenerate base-points is guaranteed by Prop. 2.8. But other properties, like being Hausdorff or even normal, are not satisfied by finite spaces (notice that every finite T1-space is discrete), so we have not explored them further. LS-CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES 7 3. LS-category of simplicial complexes We work in the category of finite simplicial complexes and simplicial maps [13]. The key notion introduced in this paper is that of LS-category in the simplicial setting. This construction is the natural one when the notion of “homotopy” is that of contiguity class. Contiguous maps were considered in Subsection 2.1. 3.1. Simplicial category. Definition 3.1. Let Kbe a simplicial complex. We say that the subcomplex U⊂Kis categorical if there exists a vertex v∈Ksuch that the inclusion i:U→Kand the constant map cv:U→Kare in the same contiguity class, iU∼cv. In other words, ifactors through vup to “homotopy” (in the sense of contiguity class). Notice that a categorical subcomplex may not be connected. Definition 3.2. The simplicial LS-category, scat K, of the simplicial complex K, is the least integer m≥0 such that Kcan be covered by m+ 1 categorical subcomplexes. For instance, scat K= 0 if and only if Khas the strong homotopy type of a point. Example 3.3. The simplicial complex Kof Figure 2 appears in [3]. It is collapsible (in the usual sense) but not strongly collapsible, so scat K≥1. We can obtain a cover by two strongly collapsible subcomplexes taking a non internal 2-simplex σand its complement K\σ. Thus scat K= 1. Figure 2. A complex Kwith scat K= 1. This example shows that scat depends on the simplicial structure more than on the geometric realization of the complex. 3.2. Homotopical invariance. The most important property of the simplicial category is that it is an invariant of the strong equivalence type, as we shall prove now. Theorem 3.4. Let K∼Lbe two strongly equivalent complexes. Then scat K= scat L. We begin with two Lemmas which are easy to prove. 8 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES Lemma 3.5. Let f, g :K→Lbe two contiguous maps, f∼cg, and let i:N→K(resp. r:L→N) be another simplicial map. Then f◦i∼cg◦i (resp. r◦f∼cr◦g). Lemma 3.6. Let K=K0 f1 →K1→ · · · fn →Kn=L and L=Kn gn → · · · → K1 g1 →K0=K be two sequences of maps such that gi◦fi∼c1and fi◦gi∼c1, for all i∈ {1, . . . , n}. Then the complexes Kand Lare strongly equivalent, K∼L. The main Theorem 3.4 will be a direct consequence of the following Proposition (by interchanging the roles of Kand L). Proposition 3.7. Let f:K→Land g:L→Kbe simplicial maps such that g◦f∼1K. Then scat K≤scat L. Proof. Let U⊂Lbe a categorical subcomplex. Since the inclusion iUis in the contiguity class of some constant map cv, there exists a sequence of maps ϕi:U→K, 0 ≤i≤n, such that iU=ϕ0∼c· · · ∼cϕn=cv. Take the subcomplex f−1(U)⊂K. We shall prove that f−1(U) is categorical. Since g◦f∼1K, there is a sequence of maps ψi:K→K, 0 ≤i≤m, such that 1K=ψ0∼c· · · ∼cψm=g◦f. Denote by f0the restriction of fto f−1(U), with values in U, that is, f0:f−1(U)→U, defined by f0(x) = f(x). Denote by j:f−1(U)⊂Kthe inclusion. Then: (1) j= 1K◦j=ψ0◦j∼c· · · ∼cψm◦j=g◦f◦j by Lemma 3.5. Since f◦j=iU◦f0, we have (2) g◦f◦j=g◦iU◦f0=g◦ϕ0◦f0∼c· · · ∼cg◦ϕn◦f0. But ϕn=cv, so g◦ϕn◦f0:f−1(U)→g(U) is the constant map cg(v). Combining (1) and (2) we obtain j∼cg(v). Therefore, the subcomplex f−1(U)⊂Kis categorical. Finally, let k= scat Land let {U0, . . . , Uk}be a categorical cover of L; then {f−1(U0), . . . , f−1(Uk)}is a categorical cover of K, which shows that scat K≤k. Acore of a finite simplicial complex Kis a subcomplex K0⊂Kwithout dominated vertices, such that K&& K0[3]. Every complex has a core, which is unique up to isomorphism, and two finite simplicial complexes have the same strong homotopy type if and only if their cores are isomorphic. LS-CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES 9 Since scat is an invariant of the strong homotopy type (Theorem 3.4) we have proved the following result. Corollary 3.8. Let K0be the core of the simplicial complex K. Then scat K= scat K0. 4. Geometric category As in the classical case, we shall introduce a notion of simplicial geometric category gscat in the simplicial setting, when “homotopy” means to be in the same contiguity class. Another so-called discrete category, dcat, which takes into account the notion of collapsibility instead of strong collapsibility, has been considered by Scoville et al in [1]. But in contrast with the simplicial LS-category introduced in Section 3, both gscat and dcat are not homotopy invariant. The problem must then be overcome by taking the infimum of the category values over all simplicial complexes which are homotopy equivalent to the given one. However, our geometric category possesses a remarkable property: due to the notion of core complex explained before, there is also a maximum of category among the complexes in a given homotopy class. Remark 3.It is possible to do a translation of the notion of simple collapsibility to finite topological spaces, by means of the notion of weak beat point [6]. 4.1. Simplicial geometric category. According to the notion of strong collapse (defined in Section 2), a simplicial complex Kis strongly collapsible if it is strongly equivalent to a point. Equivalently, the identity 1Kis in the contiguity class of some constant map cv:K→K. Definition 4.1. The simplicial geometric category gscat Kof the simplicial complex Kis the least integer m≥0 such that Kcan be covered by m+ 1 strongly collapsible subcomplexes. That is, there exists a cover U0, . . . , Um⊂ Kof Ksuch that Ui∼ ∗, for all i∈ {0,...m}. Notice that strongly collapsible subcomplexes must be connected. Proposition 4.2. scat K≤gscat K. Proof. The proof is reduced to checking that a strongly collapsible subcomplex is categorical: in fact, the only difference is that in the first case the identity 1Uis in the contiguity class of some constant map cv, while in the second it is the inclusion iU:U→Xthat satisfies iU∼cv. 4.2. Behaviour under strong collapses. Obviously, scat and gscat are invariant by simplicial isomorphisms. Moreover we proved in Theorem 3.4 that scat is a homotopy invariant. The next Theorem shows that strong collapses increase the geometric category. Theorem 4.3. If Lis a strong collapse of Kthen gscat L≥gscat K. 16 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES Figure 8. The order complex K(X) of the space Xgiven in Figure 7. Figure 9. Two strongly collapsible subcomplexes Proposition 6.4. Let Kbe a simplicial complex and χ(K)its Hasse diagram. Then cat χ(K)≤scat K. Proof. Let K0, . . . , Knbe a cover of Kby subcomplexes such that each inclusion ik:Kk⊂Kis in the same contiguity class of some constant map ck:Kk→K. Then, using Proposition 2.10, the continuous maps χ(ik) and χ(ck) are homotopic. By definition (Section 2.3), the first one is the inclusion χ(Kk)⊂χ(K), and the second one is a constant map. Then χ(K0), . . . , χ(Kn) is a categorical cover of χ(K). Thus cat χ(K)≤n. A completely analogous proof gives the corresponding result for geometric categories. Proposition 6.5. gcat χ(K)≤gscat K. The next Corollary is a direct reformulation in categorical terms of original results due to J. Barmak (Corollary 5.2.8 of [2]). Corollary 6.6. (1) cat X= 0 if and only if scat K(X)=0. In other words, Xis contractible if and only if its order complex K(X)is strongly collapsible. (2) scat K= 0 if and only if cat χ(K) = 0, that is the complex Kis strongly collapsible if and only if its order poset χ(K)is contractible. Finally, we compare the simplicial category of a complex and of its first barycentric subdivision. Corollary 6.7. If Kis a simplicial complex, then the category of its first barycentric subdivision satisfies scat sd(K)≤scat K. Proof. Since K0=K(χ(K)) equals sd(K), it follows from Propositions 6.1 and 6.4 that scat K0≤cat χ(K)≤scat K.  LS-CATEGORY OF SIMPLICIAL COMPLEXES AND FINITE SPACES 17 Notice that a complex Kand its barycentric subdivision sd(K) may not have the same strong homotopy type. For instance [2, Example 5.1.13] if Kis the boundary of a 2-simplex then both complexes Kand sd(K) do not have beat points. Then if they were in the same homotopy class they would be isomorphic, by Stong’s result [14, Th. 3]. But obviously they are not. However, as pointed out in the proof of Corollary 6.6, a complex Kis strong collapsible if and only if its barycentric subdivsion sd(K) is strong collapsible. 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McGraw-Hill Series in Higher Mathematics. New York (1966). [14] Stong, R.E. Finite topological spaces. Trans. Amer. Math. Soc. 123, 325–340 (1966). [15] Wachs, M.L. Poset Topology: tools and applications. In Geometric combinatorics 497–615, Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Studies (2007). [16] Wofsey, E. On the algebraic topology of finite spaces (2008) http://www.math. harvard.edu/~waffle/finitespaces.pdf. 18 D. FERN´ ANDEZ-TERNERO, E. MAC´ IAS-VIRG ´ OS, AND J.A. VILCHES Desamparados Fern´ andez-Ternero. Dpto. de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Spain. [email protected] Enrique Mac´ ıas-Virg´ os. Dpto. de Geometr´ıa y Topolog´ıa, Universidade de Santiago de Compostela, Spain. [email protected] Jos´ e Antonio Vilches Alarc´ on. Dpto. de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Spain. vilc[email protected]