Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Journal of Topology and Analysis (2021) ©World Scientific Publishing Company DOI: 10.1142/S1793525321500709 Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem D. Fern´andez-Ternero∗,‡,E.Mac´ıas-Virg´os†,§, D. Mosquera-Lois†,¶andJ.A.Vilches ∗, ∗ Departamento de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Spain † Instituto de Matem´aticas, Universidade de Santiago de Compostela, Spain ‡ [email protected] § [email protected] ¶ david.mosquer[email protected] [email protected] Received 16 March 2021 Revised 8 October 2021 Accepted 20 November 2021 Published 30 December 2021 We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik– Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions. Keywords: Morse–Bott functions; posets; Lusternik–Schnirelmann theorem. AMS Subject Classification: 57R70, 37B35, 55M30, 58E05, 06A07 1. Introduction Since its introduction, Morse Theory has been an active field of research with connections with many different areas of Mathematics. That interaction has led to several adaptations of Morse Theory to different contexts, for example: PL versions by Banchoff [2, 3] and by Bestvina and Brady [7] and a purely combinatorial approach by Forman [19, 20]. Nowadays, not only pure mathematics benefit from that interaction, but also applied mathematics [25] due to the importance of discrete settings. Roughly speaking, Morse Theory addresses the study of the topology (homology, originally) of a space by breaking it into “elementary” pieces. That is achieved by the so-called Fundamental or Structural Theorems of Morse Theory, which assert §Corresponding author. 1 J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 2D. Fern´andez-Ternero et al. that the object of study (for example a smooth manifold or a simplicial complex) has the homotopy type of a CW-complex with a given cell structure determined by the criticality of a Morse function defined on it [19, 33]. Originally, Morse Theory began with the definition of Morse function itself, that is, a smooth function with nondegenerate critical points. So, the only critical objects allowed were points. That was overcome by Morse–Bott theory [9], which broadened the class of critical objects by including nondegenerate critical submanifolds. The attempt to avoid the condition that the critical objects must be nondegenerate led to the introduction of Lusternik–Schnirelmann theory [30]. In the context of Morse Theory, Morse inequalities guarantee that the number of critical points of a Morse function f:X→Ris an upper bound for the homological complexity of the space X. The role of the Morse inequalities in the setting of Lusternik–Schnirelmann theory is played by the so-called Lusternik–Schnirelmann theorem, which asserts that the weighted sum of the number of critical objects is an upper bound for the category of the space [30]. Recent works have shown that it is possible to approach important problems regarding posets by using topological methods. See for example Barmak and Minian’s work on the realizability of groups as the automorphism groups of certain posets [5] or Stong’s work on groups explaining the way in that the homotopy type of the poset of nontrivial p-subgroups ordered by inclusion determines algebraic properties of the group [37]. Moreover, it is expected that recent discrete analogues of some classical concepts from differential topology will shed light on their original counterparts [22]. Therefore, it makes sense to study the topology of finite spaces by means of some version of Morse theory adapted to this context. This approach was first introduced by Minian [34] and later continued by Kukiela [29]. An invariant of a space Xis usually defined as the smallest number of open sets that cover Xand that satisfy certain properties, such as being elementary in a certain sense, for instance acyclic or contractible (see [23, 26] for more examples). More generally, and analogously, a categorical invariant of an object X(such as a simplicial complex) can be defined as the smallest number of subobjects needed to cover Xand that verify certain properties (see for example [14, 15, 31, 39]). In vague terms, such an invariant provides a certain measure of the complexity of an object. For example, the Lusternik–Schnirelmann category measures, in a particular manner, how far is a space from being contractible. This work addresses two aims. First, to develop Morse–Bott theory in the context of finite spaces, generalizing both Morse theory for posets introduced by Minian [34] and discrete Morse–Bott theory for complexes [18]. In particular, we prove: an integration result for matchings, the Fundamental Theorems of Morse–Bott theory in this setting and several generalizations of Morse inequalities for arbitrary matchings. Second, we introduce a homological LS-category and we prove the Lusternik– Schnirelmann theorem. We now describe the main motivations for the introduction of a homological LS- category. The first one is the absence of a discrete Lusternik–Schnirelmann theorem J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 3 for arbitrary matchings, not even in the simplicial setting, and second, the lack in the literature of a Lusternik–Schnirelmann theorem even for Morse (acyclic) matchings. Nevertheless, several attempts were made. On the one side, a subset of the authors in joint work with Scoville proved a Lusternik–Schnirelmann theorem for a notion of simplicial category and acyclic matchings in the context of simplicial complexes [17]. However, in order to do so, they developed another notion of criticality which leads to a different and non-equivalent definition of discrete Morse function. On the other side, first Scoville and Aaronson [1], then Tanaka [38], and afterwards Knudson and Johnson [28], approached the task by defining another categorical invariant while keeping the usual definition of discrete Morse function. The organization of the paper is as follows. In Sec. 2 we recall some definitions and standard results about posets, finite topological spaces and regular complexes. Section 3 is devoted to the study of Morse–Bott theory in the context of posets. In Sec. 4 we prove the Fundamental Theorems of Morse–Bott Theory in this setting and exploit some of their consequences. In Sec. 5 we introduce a new notion of homological LS-category and prove the corresponding Lusternik–Schnirelmann theorem for general matchings. Finally, Sec. 6 is devoted to working out a complete example to illustrate the main ideas, definitions and results in the paper. Following the suggestion of an anonymous referee, we sketch several proofs which follow closely the ones in the simplicial setting due to Forman. 2. Finite Spaces, Posets and Simplicial Complexes This section is devoted to introducing the objects we will work with. Most of the material is well established in the literature, for further details or proofs the reader is referred to [4, 6, 8, 13, 16, 34, 40]. 2.1. Finite spaces and posets It is well known that finite posets (partially ordered sets) and finite T0-spaces are in bijective correspondence. If (X,≤) is a poset, a topology Ton Xis given by taking the sets Ux:= {y∈X:y≤x} as a basis. On the other hand, if Xis a finite T0-space, define for each x∈Xthe minimal open set Uxas the intersection of all open sets containing x.ThenXmay be given an order by defining y≤xif and only if Uy⊂Ux. It is easy to see that these correspondences are mutual inverses of each other. Moreover a map between posets f:X→Yis order preserving if and only if it is continuous when considered as a map between the associated finite spaces. All posets will be assumed to be finite and by finite space we will mean T0-space. We will use the notion of finite T0-space and poset interchangeably. We need to introduce some terminology. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 4D. Fern´andez-Ternero et al. Definition 2.1. Achain in a poset Xis a subset C⊆Xsuch that if x, y ∈C, then either x≤yor y≤x. Definition 2.2. The height of a poset Xis the maximum length of the chains in X, where the chain x0<x 1<···<x nhas length n. The height h(x)ofanelement x∈Xis the height of Uxwith the induced order. Definition 2.3. AposetXis said to be homogeneous of degree nif all maximal chains in Xhave length n.Aposetisgraded if Uxis homogeneous for every x∈X. In that case, the degree of x, denoted by deg(x), is its height. We will denote both the height and degree of an element by superscripts, for example x(p). Let Xbe a finite poset, x, y ∈X.Ifx<yand there is no z∈Xsuch that x<z<y,wewritex≺y. For x∈Xwe also define  Ux:= {w∈X:w<x}as well as Fx:= {y∈X:y≥ x}and  Fx:= {y∈X:y>x}. 2.2. The McCord functors We now recall McCord functors between posets and simplicial complexes (see [32]). Given a poset X, we define its order complex K(X) as the simplicial complex whose k-simplices are the non-empty k-chains of X. Furthermore, given an order preserving map f:X→Ybetween posets, we define the simplicial map K(f): K(X)→K(Y)givenbyK(f)(x)=f(x). Conversely, if Kis a simplicial complex, we define the face poset of K,as the poset Δ(K) of simplices of Kordered by inclusion. Given a simplicial map φ:K→Lwe define the order preserving map Δ(φ): Δ(K)→Δ(L)givenby Δ(φ)(σ)=φ(σ) for each simplex σof K. The face poset functor can be defined analogously for regular CW-complexes. That is, given a regular CW-complex K,Δ(K) is the poset of cells of Kordered by inclusion. Given a cellular map φ:K→Lwe define the order preserving map Δ(φ): Δ(K)→Δ(L)givenbyΔ(φ)(σ)=φ(σ) for each cell σof K. Note that for the simplicial complex K,KΔ(K)issd(K), the first barycentric subdivision of K. By analogy, the first subdivision of a finite poset Xis defined as ΔK(X). Theorem 2.1. The following statements hold: (1) Let Xbe a finite T0-space. Then there is a map μX:|K(X)|→Xwhich is a weak homotopy equivalence. (2) Let Kbe a simplicial complex. Then there is a map μK:|K|→Δ(K)which is a weak homotopy equivalence. The maps μX:|K(X)|→Xand μK:|K|→Δ(K) will be referred as McCord maps. For details and a proof of the result above see [4]. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 5 2.3. Cellular poset homology We shall consider a special kind of posets called cellular. They were first introduced by Farmer [13] and then recovered by Minian [34]. Farmer’s definition is more general while Minian’s one is more adequate for our purposes. That is why we present the latter one. Definition 2.4 ([34]). The poset Xis cellular if it is graded and for every x∈X,  Uxhas the homology of a (p−1)-sphere, where pis the degree of x. Let Xbe a cellular poset. We denote by H∗(X) the singular homology of X. Unless stated otherwise, homology will be considered with integer coefficients. However, the constructions work as well for homology modules with coefficients in any principal ideal domain. We recall the construction due to Farmer [13] and Minian [34] of a “cellular homology theory” for cellular posets. Definition 2.5. Given a finite graded poset X, we define X(p)as the subposet of elements of degree less or equal than p, i.e. X(p)={x∈X:deg(x)≤p}. Given the cellular poset X, there is a natural filtration by the degree X(0) ⊂X(1) ⊂···⊂X(n)=X, which allows to define a cellular chain complex (C∗,d) as follows: Cp(X)=Hp(X(p),X(p−1))=  deg(x)=p Hp−1( Ux), which is a free abelian group with one generator for each element of Xof degree p. The differential d:Cp(X)→Cp−1(X) is defined as the composition Hp(X(p),X(p−1))Hp−1(X(p−1))Hp−1(X(p−1),X(p−2)), ∂j where jis the canonical map induced by the inclusion and ∂is the connecting homomorphism coming from the long exact sequence associated to the pair (X(p),X(p−1)). It can be shown (see [34]) that the differential can be written as d(x)=w≺x(x, w)wwhere the incidence number (x, w) is the degree of the map  ∂:Z=Hp−1( Ux)→Hp−2( Uw)=Z, which coincides with the connecting morphism of the Mayer–Vietoris sequence associated to the covering  Ux=( Ux−{w})∪Uw(see [34]). Theorem 2.2 ([34, Theorem 3.7]). Let Xbe a cellular poset,then H∗(C∗(X)) ∼ =H∗(X). J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 6D. Fern´andez-Ternero et al. 2.4. Homologically admissible posets We recall the notion of homologically admissible posets introduced by Minian [34]. We denote by H(X) the Hasse diagram associated to the poset X. Definition 2.6 ([34]). Let Xbe a poset. An edge (w, x)∈H(X) is homologically admissible if  Ux−{w}is acyclic. A poset is homologically admissible if all its edges are homologically admissible. The importance of homologically admissible posets lies, partially, in the following result. Lemma 2.1 ([34, Remark 3.9]). If (w,x)is a homologically admissible edge of a cellular poset X, then the incidence number (x, w)is 1or −1. Remark 2.1. The face posets of regular CW-complexes are homologically admissible [34, Remark 2.6]. However, not every homologically admissible poset is the face poset of a regular CW-complex [34, Example 2.7]. Lemma 2.2 ([34]). Let Xbe a poset. If Xis homologically admissible,then it is cellular. Remark 2.2. In Lemma 2.2 it is assumed that the empty set is not acyclic. 2.5. Euler characteristic Definition 2.7. Let Xbe a finite graded poset of degree n.DenotebyX(=p)the elements of degree pin X.Thegraded Euler-Poincar´e characteristic of Xis defined as the number: χg(X)= n  p=0 (−1)p#X(=p). It is clear that given a poset Xof the form X=Δ(K) for a finite simplicial complex K,thenχg(X)=χ(K(X)). Moreover, as a consequence of Minian’s result (Theorem 2.2), the standard homological argument (see for example [27, pp. 146– 147]) proves that for a finite cellular poset X,χg(X)=χ(K(X)). However, this does not hold in general for finite posets as the following example illustrates. Fig. 1. A poset where χg(X)=χ(K(X)). J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 7 Example 2.1. Consider the poset Xrepresented in Fig. 1. Due to the homotopic invariance of χ,χ(K(X)) = 1 because Xis contractible by removing beat points (see [4]). However, χg(X)=2. 3. Dynamics and Morse–Bott Functions for Posets In this section, we generalize the notion of discrete Morse–Bott function, introduced by Forman in the setting of complexes [18], to the context of posets. It also generalizes that of Morse functions on posets defined by Minian [34]. 3.1. Morse functions We recall the definition of Morse function for posets introduced by Minian [34]. Definition 3.1. Let Xbe a finite poset. A Morse function is a function f:X→R such that, for every x∈X,wehave #{y∈X:x≺yand f(x)≥f(y)}≤1 and #{z∈X:z≺xand f(z)≥f(x)}≤1. If fis a Morse function, an element x∈Xis said to be critical if #{y∈X:x≺yand f(x)≥f(y)}=0 and #{z∈X:z≺xand f(z)≥f(x)}=0. The set of critical points of fis denoted by critf. The images of the critical points are called critical values and the real numbers which are not critical are called regular values. The points which are not critical are said to be regular points. 3.2. Matchings Forman [21] introduced combinatorial vector fields. It is easy to see that this notion can be substituted by the concept of matching introduced to the context of discrete Morse Theory by Chari [10]. Definition 3.2. Amatching in a poset Xis a subset M⊂X×Xsuch that •(x, y)∈Mimplies x≺y; •each x∈Xbelongs to at most one element in M. Given a poset X, let us denote by H(X) its associated Hasse diagram. If M is a matching in X,writeHM(X) for the directed graph obtained from H(X)by reversing the orientations of the edges which are not in M. Any node of H(X)not J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 8D. Fern´andez-Ternero et al. incident with any edge of Mis called critical. The set of all critical nodes of Mis denoted by CM. Definition 3.3. Let Mbe a matching on a poset Xand let x(p)and ˜x(p)be two elements of X.AnM-path γof index pfrom x(p)to ˜x(p)is a sequence: γ:x=x(p) 0≺y(p+1) 0x(p) 1≺y(p+1) 1···≺y(p+1) r−1x(p) r=˜x(p) such that for each i=0,1,...,r−1 with r≥1: (1) (xi,y i)∈M, (2) xi=xi+1. AM-cycle γin HM(X) is a closed M-path in HM(X) seen as a directed graph. And the matching Mis said to be a Morse matching if HM(X) is acyclic. 3.3. Critical subposets In this subsection, we develop the notion of critical subposet (chain recurrent set) by means of matchings generalizing the analogous notion introduced by Forman [18] in the context of discrete Morse Theory. Definition 3.4. Let Mbe a matching on X.Wesaythatx(p)∈Xis an element of the chain recurrent set Rif one of the following conditions holds: •xis a critical point of M. •There is a M-cycle γin HM(X) such that x∈γ. The chain recurrent set decomposes into disjoint subsets Λiby means of the equivalence relation defined as follows: (1) If xis a critical point, then it is only related to itself. (2) Given x, y ∈R,x=y,x∼yif there is a M-cycle γsuch that x, y ∈γ. Let Λ1,...,Λkbe the equivalence classes of R.TheΛ  isare called basic sets. Each Λiconsists of either a single critical point of Mor a union of M-cycles. Example 3.1. Consider the finite model of RP2depicted in Fig. 2 (see [4, Example 7.1.1]). There is a critical point which is also a basic set, depicted with a cross. Moreover, the dashed and dotted arrows represent another two basic sets, each consisting of one cycle. Fig. 2. A finite model of RP2. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 9 3.4. Integration of matchings When working in the differentiable category, Morse theory generalizes naturally to Morse–Bott Theory. The purpose of this subsection is to generalize Minian’s integration result for matchings [34, Lemma 3.12] to the context of Morse–Bott functions and arbitrary matchings. We introduce some auxiliary notation. For each edge (x, y)∈M,wesaythatx is the source of the edge and yis the target. For convenience, we define the source and target maps (only defined for elements in the matching M) as follows: given (x, y)∈M,s(y)=xand t(x)=y. Definition 3.5. Given a matching on a finite poset Xa function f:X→Ris said to be a Morse–Bott or Lyapunov function if it is constant on each basic set and it is a Morse function away from the chain recurrent set. We say that the critical values of a Morse–Bott function are the images of the basic sets. The ideas of Forman’s proof of [18, Theorem 2.4] generalize to the context of graded posets. Theorem 3.1 (Integration of matchings). Let Xbe a finite graded poset and let Mbe a matching in X. Then there exists a Morse–Bott function f:X→R such that (1) If x(p)/∈Rand x≺y(p+1),then ⎧ ⎨ ⎩ f(x)<f(y)if (x, y)/∈M, f(x)≥f(y)if (x, y)∈M. (2) If x(p)∈Rand x≺y(p+1),then ⎧ ⎨ ⎩ f(x)=f(y)if x∼y, f(x)<f(y)if xy. Proof. First of all, we extend the equivalence relation ∼to all of Xas follows: if x/∈R,then{x}is an equivalence class. Second, we define an auxiliary map d:X→Ngiven by d(x)=max{s:∃M-path γ:x=x(p) 0≺y(p+1) 0x(p) 1≺y(p+1) 1···≺y(p+1) r−1x(p) r=˜x(p) such that the x isin γinclude elements from exactly sdistinct equivalence classes}. (1) J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 16 D. Fern´andez-Ternero et al. Remark 4.1. Observe that the multiplicity of a path is always 1 or −1 due to Lemma 2.1. The generalization of [18, Lemma 4.6] to our context is straightforward. As a consequence, [18, Theorem 7.3] generalizes to our setting with the same proof. Theorem 4.5. Let Xbe a homologically admissible poset and let Mbe a Morse– Smale matching on X. Let the coefficients ring Rbe a principal ideal domain. Let λidenote a basic set consisting of a single closed orbit [γ]of index p.Then Hk(¯ Λi,˙ Λi)∼ =0for k=p, p +1, Hp(¯ Λi,˙ Λi)∼ =⎧ ⎨ ⎩ Rif m(γ)=1, R 2Rif m(γ)=−1, Hp+1(¯ Λi,˙ Λi)∼ =⎧ ⎨ ⎩ Rif m(γ)=1, 0if m(γ)=−1. Therefore, combining the Strong Morse–Bott inequalities (Theorem 4.3) with Theorem 4.5 and taking into account that fields have no torsion, we obtain a generalization of [18, Corollary 7.4] to our setting. Theorem 4.6. Let Xbe a homologically admissible poset and let Mbe a Morse– Smale matching on X. Let the coefficients ring be the field R.DenotebyA pthe number of closed M-orbits of index pand multiplicity 1.Then,for every k≥0: A k+ k  i=0 (−1)ick−i≥ k  i=0 (−1)ibk−i(R). Remark 4.2. While [18, Corollary 7.4] refined [18, Theorem 7.2], Theorem 4.6 does not refine our improved Theorem 4.4. They are complementary results. 5. Homological Lusternik–Schnirelmann Theorem The purpose of this section is to prove a Lusternik–Schnirelmann theorem for general matchings and a suitable definition of homological category. 5.1. Homological chain category Let (C∗,∂) denote a free chain complex of abelian groups. It is bounded if only finitely many of the Cpare nonzero. Moreover, if each term Cpis finitely generated, then we define the rank of C∗as rank(C∗)=prank (Cp). Definition 5.1. Let (C∗,∂) be a free chain complex of abelian groups. We define its homological chain category hccat(C∗) = inf rank (B∗): B∗bounded subcomplex of C∗and the inclusion i:B∗→C∗is a quasi-isomorphism . J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 17 Let Xbe a topological space. We denote by S∗(X) its singular chain complex. For all the definitions that follow we consider coefficients in Z. Definition 5.2. Let Xbe a topological space. We define its homological chain category hccat(X) = hccat(S∗(X)). We introduce a homological lower bound for hccat(X) analogous to the Pitcher strengthening of Morse inequalities [16, Corollary 5.2.3]. Recall from Eq. (2) the definition of ηk. Proposition 5.1. Let Xbe a topological space with finitely generated homology. Then  k bk+2 k ηk≤hccat(X). Proof. Let us denote by (B∗,∂) a bounded chain complex whose homology is isomorphic to H∗(X). By standard algebra (see, for example [36, Theorem 4.11]), we have bk+ηk+ηk−1≤rank(Bk). Now the result follows by a sum indexed by the dimension. Corollary 5.1. Let Xbe a homologically admissible poset or a CW-complex with finitely generated homology. Then χ(X)≤hccat(X). In fact, the bound given by Proposition 5.1 is the best possible as a consequence of the following result due to Pitcher [35, Lemma 13.2]. Proposition 5.2. Let (C∗,∂)be a free chain complex with singular homology groups Hk(X),k=0,1,.... Then there exists a free chain complex (L, ∂L)such that (1) For every k≥0,the group Lkhas rank bk+ηk+ηk−1. (2) There exists a monomorphism i:L→Cwhich is a chain map. (3) The monomorphism i:L→Cis a quasi-isomorphism. Corollary 5.2. Let Xbe a topological space with finitely generated homology. Then hccat(X)= k bk+2 k ηk. Moreover, observe that a topological Xis acyclic if and only if hccat(X)=1. As a consequence of [12, Example 1.33] we have the following result relating the homological chain category to the Lusternik–Schnirelmann category. Proposition 5.3. Let Kbe a simply connected CW-complex with finitely generated homology groups such that there exists n∈Nsatisfying Hn(K)=0and Hp(K)=0 for p>n.Then cat(K)≤hccat(K). J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 18 D. Fern´andez-Ternero et al. The result does not necessarily hold if we remove the simply connectedness hypothesis, as the following example shows. Example 5.1. Consider the Poincar´e homology 3-sphere M.Observethat hccat(M) = hccat(S3)=2.However,cat(M)≥3 [23]. 5.2. Homological Lusternik–Schnirelmann theorem In this subsection, we state and prove a Lusternik–Schnirelmann theorem for the homological chain category and general matchings on posets. Theorem 5.1. Let Xbe a homologically admissible poset and let Mbe a Morse– Smale matching on X.Then hccat(X)≤ basic sets Λi hccat(Λi). In particular,if Mis a Morse matching on X, then hccat(X)is a lower bound for the number of critical elements of M. Proof. We will define another Morse matching M∗by perturbing M. The idea is to replace each prime closed orbit by two critical points. This will be achieved by removing exactly one of the edges of the matching in each closed orbit. By repeating the technique used in the proof of Theorem 4.4, we obtain a Morse matching M∗ satisfying m∗ p=cp+Ap+Ap−1,wherem∗ pdenotes the number of critical points of index pof the matching M∗(see Subsection 4.3 for the definition of Ap). Recall that C∗(X) denotes the cellular chain complex of X. We define a map V:Cp(X)→Cp+1(X) as follows: V(x)=⎧ ⎨ ⎩ −(y,x)yif there exists y∈Xwith (x, y)∈M ∗, 0otherwise. Following the ideas of Minian [34], define the discrete flow operator φ:Cp(X)→ Cp(X)asφ=id+dV +Vd.Theφ-invariant chains Cφ p(X)={c∈Cp(X): φ(c)=c} form a well-defined subcomplex of (C∗(X),d) [34]. Moreover, the inclusion of (Cφ ∗(X),d)into(C∗(X),d) induces isomorphisms in homology and Cφ p(X)isisomorphic to the free abelian group spanned by the critical p-elements of X[34]. As a consequence hccat(C∗(X)) ≤ p m∗ p= p cp+Ap+Ap−1.(3) There are two kinds of basic sets for M: critical points and disjoint closed M-orbits. Observe that if Λiis a critical point, then hccat(Λi) = 1 while if Λiis a closed orbit, J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 19 then hccat(Λi) = 2. So, from Eq. (3), it follows that hccat(C∗(X)) ≤ basic sets Λi hccat(Λi). Finally, observe that hccat(X) = hccat(C∗(X)) due to the isomorphism between cellular homology and singular homology for cellular posets (Theorem 2.2). Remark 5.1. In the proof of Theorem 5.1, Eq. (3) could also be derived as a consequence of combining our Pitcher strengthening of Morse-inequalities [16, Corollary 5.2.3] applied to the matching M∗with Corollary 5.2. As a consequence of [16, Theorem 3.3.6], we obtain the following corollary. Corollary 5.3. Let Xbe a homologically admissible poset and let f:X→Rbe a Morse function. Then hccat(X)is a lower bound for the number of critical points of f. Remark 5.2. Let Kbe a simplicial complex or, more generally, a regular CW-complex K. Recall that its face poset Δ(K) is a homologically admissible poset. Moreover, the chain complex C•(Δ(K),d)wheredis the cellular boundary operator coincides with the chain complex C•(K, ∂)where∂is the cellular -or simplicial in case Kis a simplicial complexboundary operator. Therefore, hccat(Δ(K)) = hccat(K). Hence, we have in particular a simplicial homological Lusternik–Schnirelmann theorem. 6. A Worked Out Example In this section, we work out a full example to illustrate the main ideas in the paper. 6.1. Morse–Smale matching on a homologically admissible poset Consider the homologically admissible poset depicted in Fig. 4. We will denote it by X. Moreover, in Fig. 4 we also exhibit a matching Mon X. The crosses represent critical points, the dashed edges with the circles represent a periodic orbit, and the arrows represent the matched elements. Observe that the matching Mis Morse– Smale since it only has a prime orbit, therefore disjoint from the others, and three critical points (see Definition 3.6). Fig. 4. The homologically admissible poset Xwith a matching. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 20 D. Fern´andez-Ternero et al. 6.2. Integration of the matching We proceed to illustrate how to integrate the matching Mon Xfollowing the proof of Theorem 3.1. First, in Fig. 5 we show the values of the map d:X→Ngiven by Eq. (1). Observe that d(x) is greater than one only for x∈s(M). We compute D= maxx∈Xd(x) = 5. Now, we will build a Morse–Bott or Lyapunov function f:X→ R. Recall that for x(p)∈X, we define f(x) following (F1), (F2) and (F3) in p. 10. We show the values of the Morse–Bott function f:X→Rin Fig. 6. 6.3. Fundamental theorems Recall that given the poset Xand the Morse–Bott function f:X→R,foreach t∈Rwe have the level subposet Xt= f(x)≤t Ux. We will illustrate the structural theorems in this context (Theorems 4.1 and 4.2), which describe the changes in the level subposets Xtas t∈Rincreases. We begin the analysis of the level subposets. First, the minimum value of fis attached at a critical element (see Fig. 7(a)). As we reach the value t=3/10, the inclusion i:X0→X3/10 induces an isomorphism in homology. Observe that X3/10 is acyclic (see Fig. 7(b)). Next, we reach the value t=1/2 (see Fig. 8(a)). The inclusion i:X3/10 →X1/2 induces an isomorphism in homology and X1/2is still acyclic. Fig. 5. Values of the map d:X→N. Fig. 6. Values of the map f:X→R. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 Morse–Bott theory on posets 21 Fig. 7. Level subposets X0and X3/10. Fig. 8. Level subposets X1/2and X1. Fig. 9. Level subposet X1+1/5. As we reach the value t= 1 (see Fig. 8(b)), which is critical, the inclusion i:X1/2→X1no longer induces an isomorphism in homology. The level subposet X1has the same homology as S1. The next value is t=1+1/5 (see Fig. 9). Despite corresponding to a basic set, the homology does not change. Finally,wereachthevaluet= 2 (see Fig. 6), which is critical and produces a change in the homology since X2=Xhas homology: H0(X;Z)∼ =Z,H1(X;Z)∼ =Z2 and Hk(X;Z)∼ =0fork≥2. 6.4. Homological inequalities In this last subsection, we provide the explicit computations which are necessary to check the homological inequalities (Theorems 4.3, 4.4 and 5.1). First of all, hccat(X) = 3. We provide the remaining information in tables (Tables 1, 2 and 3) J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. Re-use and distribution is strictly not permitted, except for Open Access articles. 2nd Reading December 28, 2021 18:54 WSPC/243-JTA 2150070 22 D. Fern´andez-Ternero et al. Table 1. Computation of Hk(¯ Λi,˙ Λi). Hk(¯ Λi,˙ Λi)012k≥3 Λ0Z00 0 Λ10Z00 Λ20ZZ 0 Λ300Z0 Table 2. Computation of Hk(Λi). Hk(Λi)012k≥3 Λ0Z00 0 Λ1Z00 0 Λ2ZZ00 Λ3Z00 0 Table 3. Computation of several homological and combinatorial invariants. 0123i≥4 βi1000 0 ηi0100 0 ci1110 0 Ai0100 0 mi1310 0 hccat(Λi)1112 1 and assuming we work with integer coefficients. We introduce some notation regarding the basic sets: Λ0denotes the critical point of degree zero, Λ1denotes the critical point of degree one, Λ2denotes the orbit and Λ3denotes the critical point of degree two. Acknowledgments The first and the fourth authors were partially supported by MINECO Spain Research Project MTM2015-65397-P and Junta de Andaluc´ıa Research Groups FQM-326 and FQM-189. The second author was partially supported by MINECOFEDER research project MTM2016-78647-P. The third author was partly supported by Ministerio de Ciencia, Innovaci´on y Universidades, grant FPU17/03443. The second and third authors were partially supported by Xunta de Galicia ED431C 2019/10 with FEDER funds. All the authors were partially supported by MINECO Spain Research Project PID2020 114474GB-100. References 1. S. Aaronson and N. A. Scoville, Lusternik-Schnirelmann category for simplicial complexes, Illinois J. Math. 57(3) (2013) 743–753. 2. T. F. Banchoff, Critical points and curvature for embedded polyhedra, J. Differ. Geom. 1(1967) 245–256. J. Topol. Anal. Downloaded from www.worldscientific.com by UNIVERSITY OF SEVILLE on 06/30/22. 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