Morse-Bott Theory on posets and an homological Lusternik-Schnirelmann Theorem
D. Fernández-Ternero, E. Macías-Virgós, D. Mosquera-Lois, J.A. Vilches
aa r X i v : . [ m a t h . A T ] J u l MORSE-BOTT THEORY ON POSETS AND ANHOMOLOGICAL LUSTERNIK-SCHNIRELMANNTHEOREM
D. FERN ´ANDEZ-TERNERO (1) , E. MAC´IAS-VIRG ´OS (2) ,D. MOSQUERA-LOIS (2) , J.A. VILCHES (1)
Abstract.
We develop Morse-Bott theory on posets, generalizingboth discrete Morse-Bott theory for regular complexes and Morsetheory on posets. Moreover, we prove a Lusternik-Schnirelmanntheorem for general matchings on posets, in particular, for Morse-Bott functions.
Contents
1. Introduction 12. Finite Spaces, Posets and Simplicial Complexes 33. Dynamics and Morse-Bott functions for posets 74. Fundamental Theorems and consequences 105. Homological Lusternik-Schnirelmann Theorem 15References 181.
Introduction
Since its introduction, Morse Theory has been an active field of re-search and connections with many different areas of Mathematics havebeen found. That interaction has led to several adaptations of MorseTheory to different contexts, for example: PL versions by Banchoff[2, 3] and by Bestvina and Brady [7] and a purely combinatorial ap-proach by Forman [20, 18]. Nowadays, not only pure mathematics
Mathematics Subject Classification.
Primary: 57R70 Secondary: 37B35.The first and the fourth authors were partially supported by MINECO SpainResearch Project MTM2015–65397–P and Junta de Andaluc´ıa Research GroupsFQM–326 and FQM–189. The second author was partially supported by MINECO-FEDER research project MTM2016–78647–P. The third author was partly sup-ported by Ministerio de Ciencia, Innovaci´on y Universidades, grant FPU17/03443.The second and third authors were partially supported by Xunta de Galicia ED431C2019/10 with FEDER funds. benefit from that interaction, but also applied mathematics [25] due tothe importance of discrete settings.Roughly speaking, Morse Theory addresses the study of the topol-ogy (homology, originally) of a space by breaking it into “elementary”pieces. That is achieved by the so called Fundamental or StructuralTheorems of Morse Theory, which assert that the object of study (forexample a smooth manifold or a simplicial complex) has the homotopytype of a CW-complex with a given cell structure determined by thecriticality of a Morse function defined on it [18, 32].Originally Morse Theory began with the definition of Morse functionitself, that is, a smooth function with non-degenerate critical points so,the only critical objects allowed were points. That was overcame byMorse-Bott theory [9], which broadened the class of critical objectsby including non-degenerate critical submanifolds. To avoid the crit-ical objects must be non-degenerate, this led to the introduction ofLusternik-Schnirelmann theory [29].In the context of Morse Theory, Morse inequalities guarantee thatthe number of critical points of a Morse function f : X → R is an upperbound for the homological complexity of the space X . The role of theMorse inequalities in the setting of Lusternik-Schnirelmann theory isplayed by the so called Lusternik-Schnirelmann theorem, which assertsthat the weighted sum of the number of critical objects is an upperbound for the category of the space [29].Recent works have shown that it is possible to approach importantproblems regarding posets by using topological methods. See for ex-ample Barmak and Minian’s work on the realizability of groups as theautomorphism groups of certain posets [5] or Stong’s work on groupson the way the homotopy type of the poset of non-trivial p -subgroupsordered by inclusion determines algebraic properties of the group [36].Moreover, it is expected that recent discrete analogues of some classicconcepts from differential topology shed light on their originals coun-terparts [22]. Therefore, it makes sense to study the topology of finitespaces by means of some adapted version of Morse theory to this con-text.An invariant of a space X is the smallest number of open sets thatcover X and that satisfy certain properties, such as: being elementaryin a certain sense, for instance acyclic or contractible (see [23, 26] formore examples). More generally, and analogously, a categorical invari-ant of an object X (such as a simplicial complex) can be defined asthe smallest number of subobjects needed to cover X and that verifycertain properties (see for example [30, 15, 14, 38]). In vague terms, ORSE-BOTT THEORY ON POSETS 3 an invariant provides a certain measure for the complexity of an ob-ject. For example, the Lusternik-Schnirelmann category measures, ina particular manner, how far is a space from being contractible.This work addresses two aims. First, to develop Morse-Bott the-ory in the context of finite spaces, generalizing both Morse theory forposets introduced by Minian [33] and discrete Morse-Bott theory [19].In particular, we prove an integration result for matchings, the Fun-damental Theorems of Morse-Bott theory in this setting and severalgeneralization of Morse inequalities for arbitrary matchings. Second,we introduce an homological category and we prove the Lusternik-Schnirelmann theorem.We describe the main motivations for the latter goal. First, theabsence of a discrete Lusternik-Schnirelmann theorem for arbitrarymatchings, not even in the simplicial setting, and second, the gap inthe literature even for a Lusternik-Schnirelmann theorem for Morse(acyclic) matchings. Nevertheless, several attempts were made. On theone side, a subset of the authors in joint work with Scoville proved aLusternik-Schnirelmann theorem for a notion of simplicial category andacyclic matchings in the context of simplicial complexes [17]. However,in order to do so, they developed another notion of criticality whichleads to a different and non-equivalent definition of discrete Morse func-tion. On the other side, first Scoville and Aaronson [1], then Tanaka[37], and afterwards Knudson and Johnson [28], approached the task bydefining another categorical invariant but keeping the usual definitionof discrete Morse function.The organization of the paper is as follows. In Section 2 we recallsome definitions and standard results about posets, finite topologicalspaces and regular complexes. Section 3 is devoted to the study ofMorse-Bott theory in the context of posets. In Section 4 we prove theFundamental Theorems of Morse-Bott Theory in this setting and ex-ploit some of their consequences. Finally, in Section 5 we introducea new notion of homological category and prove the correspondingLusternik-Schnirelmann theorem for general matchings.2.
Finite Spaces, Posets and Simplicial Complexes
This section is devoted to introduce the objects we will work with.Most of the material is well established in the literature, for furtherdetails or proofs the reader is referred to [4, 6, 8, 13, 16, 33, 39].2.1.
Finite spaces and posets.
It is well known that finite posetsand finite T -spaces are in bijective correspondence. If ( X, ≤ ) is a FERN ´ANDEZ-TERNERO, MAC´IAS-VIRG ´OS, MOSQUERA-LOIS, VILCHES poset, a topology T on X is given by taking the sets U x := { y ∈ X : y ≤ x } as a basis. On the other hand, if X is a finite T -space, define for each x ∈ X the minimal open set U x as the intersection of all open setscontaining x . Then X may be given an order by defining y ≤ x if andonly if U y ⊂ U x . It is easy to see that these correspondences are mutualinverses of each other. Moreover a map between posets f : X → Y isorder preserving if and only if it is continuous when considered as amap between the associated finite spaces. All posets will be assumedto be finite and by finite space we will mean T -space. We will use thenotion of finite ( T -)space and poset interchangeably.We need to introduce some terminology. Definition 2.1.1. A chain in a poset X is a subset C ⊆ X such thatif x, y ∈ C , then either x ≤ y or y ≤ x . Definition 2.1.2.
The height of a poset X is the maximum length ofthe chains in X , where the chain x < x < . . . < x n has length n . Theheight h ( x ) of an element x ∈ X is the height of U x with the inducedorder. Definition 2.1.3.
A poset X is said to be homogeneous of degree n if all maximal chains in X have length n . A poset is graded if U x ishomogeneous for every x ∈ X . In that case, the degree of x , denotedby deg( x ), is its height.We will denote both the height and degree of an element by super-scripts, for example x ( p ) .Let X be a finite poset, x, y ∈ X . If x < y and there is no z ∈ X such that x < z < y , we write x ≺ y .For x ∈ X we also define b U x := { w ∈ X : w < x } as well as F x := { y ∈ X : y ≥ x } and b F x := { y ∈ X : y > x } .2.2. The McCord functors.
We now recall McCord functors be-tween posets and simplicial complexes [31]. Given a poset X , we defineits order complex K ( X ) as the simplicial complex whose k -simplicesare the non-empty k -chains of X . Furthermore, given an order pre-serving map f : X → Y between posets, we define the simplicial map K ( f ) : K ( X ) → K ( Y ) given by K ( f )( x ) = f ( x ).Conversely, if K is a simplicial complex, we define the face posetof K , ∆( K ), as the poset of simplices of K ordered by inclusion.Given a simplicial map φ : K → L we define the order preserving map∆( φ ) : ∆( K ) → ∆( L ) given by ∆( φ )( σ ) = φ ( σ ) for each simplex σ of K . ORSE-BOTT THEORY ON POSETS 5
The face poset functor can be defined analogously for regular CW-complexes. That is, given a regular CW-complex K , ∆( K ) is the posetof cells of K ordered by inclusion. Given a cellular map φ : K → L we define the order preserving map ∆( φ ) : ∆( K ) → ∆( L ) given by∆( φ )( σ ) = φ ( σ ) for each cell σ of K .Note that for the simplicial complex K , K ∆( K ) is sd( K ), the firstbarycentric subdivision of K . By analogy, the first subdivision of afinite poset X is defined as ∆ K ( X ). Theorem 2.2.1.
The following statements hold: (1)
Let X be a finite T -space. Then there is a map µ X : |K ( X ) | → X which is a weak homotopy equivalence. (2) Let K be a simplicial complex. Then there is a map µ K : | K | → ∆( K ) which is a weak homotopy equivalence. The maps µ X : |K ( X ) | → X and µ K : | K | → ∆( K ) will be referredas McCord maps. For details and a proof of the result above see [4].2.3. Cellular poset homology.
We shall consider a special kind ofposets called cellular. They were first introduced by Farmer [13] andthen recovered by Minian [33]. Farmer’s definition is more generalwhile Minian’s one is more adequate for our purposes. That is why wepresent the latter one.
Definition 2.3.1 ([33]) . The poset X is cellular if it is graded and forevery x ∈ X , b U x has the homology of a ( p − p is thedegree of x .Let X be a cellular poset. We denote by H ∗ ( X ) the singular ho-mology of X . Unless stated otherwise, homology will be consideredwith integers coefficients. However, the constructions work as well forhomology modules with coefficients in any principal ideal domain. Werecall the construction due to Farmer [13] and Minian [33] of a “cellularhomology theory” for cellular posets. Definition 2.3.2.
Given a finite graded poset X , we define X ( p ) asthe subposet of elements of degree less or equal to 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: C p ( X ) = H p ( X ( p ) , X ( p − ) = M deg( x )= p H p − ( b U x ) , FERN ´ANDEZ-TERNERO, MAC´IAS-VIRG ´OS, MOSQUERA-LOIS, VILCHES which is a free abelian group with one generator for each element of X of degree p . The differential d : C p ( X ) → C p − ( X ) is defined as thecomposition H p ( X ( p ) , X ( p − ) H p − ( X ( p − ) H p − ( X ( p − , X ( p − ) ∂ j where j is the canonical map induced by the inclusion and ∂ is theconecting homomorphism coming from the long exact sequence associ-ated to the pair ( X ( p ) , X ( p − ). It can be shown [33] that the differential d : C p ( X ) → C p − ( X )can be written as d ( x ) = P w ≺ x ǫ ( x, w ) w where the incidence number ǫ ( x, w ) is the degree of the map e ∂ : Z = H p − ( b U x ) → H p − ( b U w ) = Z . which coincides with the connecting morphism of the Mayer-Vietorissequence associated to the covering b U x = ( b U x − { w } ) ∪ U w [33]. Theorem 2.3.3 ([33, Theorem 3.7]) . Let X be a cellular poset, then H ∗ ( C ∗ ( X )) ∼ = H ∗ ( X ) . Homologically admissible posets.
We recall the notion of ho-mologically admissible posets introduced by Minian [33]. We denoteby H ( X ) the Hasse diagram associated to the poset X . Definition 2.4.1 ([33]) . Let X be a poset. An edge ( w, x ) ∈ H ( X ) ishomologically admissible if b U x −{ w } is acyclic. A poset is homologicallyadmissible if all its edges are homologically admissible.The importance of homologically admissible posets, lies, partially, inthe following result. Lemma 2.4.2 ([33, Remark 3.9]) . If ( w, x ) is a homologically admis-sible edge of a cellular poset X , then the incidence number ǫ ( x, w ) is or − .Remark . The face posets of regular CW-complexes are homologicallyadmissible [33, Remark 2.6]. However, not every homologically admis-sible poset is the face poset of a regular CW-complex [33, Example2.7].
Lemma 2.4.3 ([33]) . Let X be a poset. If X is homologically admis-sible, then it is cellular.Remark . In Lemma 2.4.3 it is assumed that the empty set is notacyclic.
ORSE-BOTT THEORY ON POSETS 7
Euler Characteristic.Definition 2.5.1.
Let X be a finite graded poset of degree n . De-note by X (= p ) the elements of degree p . The graded Euler-Poincar´echaracteristic of X is defined as the number: χ g ( X ) = n X i =0 ( − i X (= p ) . It is clear that given a poset X of the form X = ∆( K ) for a finitesimplicial complex K , then χ g ( X ) = χ ( K ( X )). Moreover, as a conse-quence of Minian’s result (Theorem 2.3.3), the standard homologicalargument (see for example [27, p. 146-147]) proves that for a finitecellular poset X , χ g ( X ) = χ ( K ( X )). However, this does not hold ingeneral for finite posets as the following example illustrates: Example 2.5.2.
Consider the poset X represented in Figure 2.1. Dueto the homotopic invariance of χ , χ ( K ( X )) = 1 because X is con-tractible by removing beat points. However, χ g ( X ) = 2. Figure 2.1.
A poset where χ g ( X ) = χ ( K ( X )).3. Dynamics and Morse-Bott functions for posets
In this section we generalize the notion of discrete Morse-Bott func-tion, introduced by Forman [19], to the context of posets. This alsogeneralizes that of Morse functions on posets defined by Minian [33].3.1.
Morse functions.
We recall the definition of Morse function forposets introduced by Minian [33].
Definition 3.1.1.
Let X be a finite poset. A Morse function is afunction f : X → R such that, for every x ∈ X , we have { y ∈ X : x ≺ y and f ( x ) ≥ f ( y ) } ≤ { z ∈ X : z ≺ x and f ( z ) ≥ f ( x ) } ≤ . If f is a Morse function, an element x ∈ X is said to be critical if { y ∈ X : x ≺ y and f ( x ) ≥ f ( y ) } = 0 FERN ´ANDEZ-TERNERO, MAC´IAS-VIRG ´OS, MOSQUERA-LOIS, VILCHES and { z ∈ X : z ≺ x and f ( z ) ≥ f ( x ) } = 0 . The set of critical points is denoted by crit f . The images of thecritical points are called critical values and the real numbers which arenot critical are called regular values . The points which are not criticalvalues 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 ofmatching introduced to the context of discrete Morse Theory by Chari[10].
Definition 3.2.1. A matching in a poset X is a subset M ⊂ X × X such that • ( x, y ) ∈ M implies x ≺ y ; • each x ∈ X belongs to at most one element in M .Given a poset X , let us denote by H ( X ) its associated Hasse dia-gram. If M is a matching in X , write H M ( X ) for the directed graphobtained from H ( X ) by reversing the orientations of the edges whichare not in M . Any node of H ( X ) not incident with any edge of M iscalled critical . The set of all critical nodes of M is denoted by C M . Definition 3.2.2.
Let M be a matching on a poset X and let x ( p ) and˜ x ( p ) be two elements of X . An M -path, γ , of index p from 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 − ≻ x ( p ) r = ˜ x ( p ) such that for each i = 0 , , . . . , r − r ≥ x i , y i ) ∈ M ,(2) x i = x i +1 .A M -cycle γ in H M ( X ) is a closed M -path in H M ( X ) seen as adirected graph. And the matching M is said to be a Morse matching if H M ( X ) is acyclic.3.3. Critical subposets.
In this subsection we develop the notion ofcritical subposet ( chain recurrent set ) by means of matchings general-izing the analogous notion introduced by Forman [19] in the context ofdiscrete Morse Theory.
Definition 3.3.1.
Let M be a matching on X . We say that x ( p ) ∈ X isan element of the chain recurrent set R if one of the following conditionsholds: ORSE-BOTT THEORY ON POSETS 9 • x is a critical point of M . • There is a M -cycle γ in H M ( X ) such that x ∈ γ .The chain recurrent set decomposes into disjoint subsets Λ i by meansof the equivalence relation defined as follows:(1) If x is a critical point, then it is only related to itself.(2) Given x, y ∈ R , x = y , x ∼ y if there is cycle γ such that x, y ∈ γ .Let Λ , . . . , Λ k be the equivalence classes of R . The Λ ′ i s are called basic sets . Each Λ i consists of either a single critical point of M or aunion of cycles. Example 3.3.2.
Consider the finite model of R P depicted in Figure3.1 (see [4, Example 7.1.1]). There is a critical point which is alsoa basic set, depicted with a cross. Moreover, the dashed and dottedarrows represent another two basic sets, each consisting of one cycle. Figure 3.1.
A finite model of R P .3.4. Integration of matchings.
When working on the differentiablecategory, Morse theory generalizes naturally to Morse-Bott Theory.The purpose of this subsection is to generalize Minian’s integrationresult for matchings [33, Lemma 3.12] to the context of Morse-Bottfunctions and arbitrary matchings.
Definition 3.4.1.
Given a matching on a finite poset X a function f : X → R is said to be a Morse-Bott or Lyapunov function if it isconstant on each basic set and it is a Morse function away from thechain recurrent set.We say that the critical values of a Morse-Bott function are theimages of the basic sets. The ideas of Forman’s proof of [19, Theorem2.4] generalize to the context of graded posets giving:
Theorem 3.4.2 (Integration of matchings) . Let X be a finite gradedposet and let M be a matching in X . Then there exists a Morse-Bottfunction f : X → R such that: (1) If x ( p ) / ∈ R and y ( p +1) ≻ x , then ( f ( x ) < f ( y ) , if ( x, y ) / ∈ M ,f ( x ) ≥ f ( y ) , if ( x, y ) ∈ M . (2) If x ( p ) ∈ R and y ( p +1) ≻ x , then ( f ( x ) = f ( y ) , if x ∼ y,f ( x ) < f ( y ) , if x ≁ y. We introduce the following definition: given a finite poset X and aMorse-Bott function f : X → R , for each a ∈ R we write X a = [ f ( x ) ≤ a U x . Morse-Smale matchings.
In this subsection we generalize thenotion of Morse-Smale vector field from the context of simplicial com-plexes [19] to the setting of finite spaces.Let X be a homologically admissible poset and let M be a matchingon X . A M -cycle γ is prime if they do not exist a natural number n > M -cycle e γ such that γ is the concatenation of e γ n times(see [19, Definition 5.3] for details).An equivalence relation on the set of M -cycles is defined as follows.Two M -cycles γ and e γ are equivalent if e γ is the result of varying thestarting point of γ (see [19, p. 631] for an example). An equivalenceclass of M -cycles is called a closed M -orbit . The equivalence class of γ is denoted by [ γ ]. The concepts of prime closed M -orbit and indexof an closed M -orbit are defined as expected (see [19] for details).A special kind of matching which will play an important role is thefollowing. In a certain sense, we control the complexity of the chainrecurrent set. Definition 3.5.1.
Let X be a homologically admissible poset. Amatching M on X is a Morse-Smale matching if the chain recurrentset R consists only of critical points and pairwise disjoint prime closed M -orbits.4. Fundamental Theorems and consequences
The purpose of this Section is to prove the Fundamental Theoremsof Morse Theory for Morse-Bott functions on posets and obtain someconsequences.
ORSE-BOTT THEORY ON POSETS 11
Fundamental Theorems.
In what follows, we extend the equiv-alence relation defined in Subsection 3.3 from R to all X by saying thata point which is not critical is an equivalence class on its own. Definition 4.1.1.
Given a finite poset X , x ∈ X and a matching M on X , we define: ∂ [ x ] = { w ∈ X : w ≺ ˜ x for some ˜ x ∼ x but w ≁ ˜ x } . Example 4.1.2.
Consider the poset depicted in Figure 3.1. In Figure4.1 we show ∂ [ x ] for any x in the dashed cycle of Figure 3.1. Figure 4.1.
Example of ∂ [ x ].We introduce some auxiliary notation. For each edge ( x, y ) ∈ M , wesay that x is the source of the edge and y is the target . For convenience,we define the source and target maps (only defined for elements in thematching M ) as follows: given ( x, y ) ∈ M , s ( y ) = x and t ( x ) = y .The lemma below follows from the definition of matching: Lemma 4.1.3.
Let γ be a cycle of index p and let u ( p − ∈ X , ˜ v ( p ) ∈ X , w ( p +1) ∈ X and r ( p +2) ∈ X such that u, ˜ v, w, r / ∈ γ . Then it holds thefollowing: t ( u ) / ∈ γ, t (˜ v ) / ∈ γ, s ( w ) / ∈ γ and s ( r ) / ∈ γ. Our next result is a homological collapsing theorem for Morse-Bottfunctions. As a consequence of the Lemma 4.1.3, the elements of acycle can not be connected by arrows with elements which are not inthe cycle. Therefore, the result below follows from [16, Theorem 4.2.2].
Theorem 4.1.4.
Let X be a finite homologically admissible poset andlet f : X → R be a Morse-Bott function. If [ a, b ] contains no criticalvalues, then i : X a ֒ → X b induces an isomorphism in homology. In this generalized context, we also have a result which explains whathappens when a critical value is reached.
Theorem 4.1.5.
Let X be a finite homologically admissible poset andlet f : X → R be a Morse-Bott function. If f ( x ) ∈ [ a, b ] is a criticalvalue and there are no other values of f in [ a, b ] , then X b = X a ∪ ∂ [ x ] [ x ] . Proof.
There are two cases to consider. First, assume that [ x ] is acritical point, then the results reduces to [16, Theorem 4.2.8]. So,assume [ x ] is a cycle of index p . Let ˜ f : X/ ∼→ R denote the functioninduced by f on the set of equivalence classes. We may assume that˜ f is injective, that ˜ f ([ x ]) > a and that the only critical subposet in f − ([ a, b ]) is [ x ].Since [ x ] is a cycle and f ( x ) is a critical value, then given y ( p +1) ≻ ˜ x and y / ∈ [ x ] with ˜ x ∈ [ x ], f ( y ) > f (˜ x ). Hence, f ( y ) > b and Lemma4.1.3 guarantees that f ( z ) > b for every z > ˜ x , z / ∈ [ x ]. Therefore,[ x ] ∩ X a = ∅ . Given any w ( p − ≺ ˜ x ( p ) , ˜ x ∈ [ x ] and w / ∈ [ x ] or w ( p ) ≺ ˜ x ( p +1) , ˜ x ∈ [ x ] and w / ∈ [ x ], due to the criticality of [ x ], it holds that f ( w ) < f (˜ x ). Therefore f ( w ) < a and w ∈ X a . Hence ∂ [ x ] ⊂ X a .That is, X b = X a ∪ ∂ [ x ] [ x ]. (cid:3) Morse-Bott inequalities.
In this subsection we generalize Morse-Bott inequalities from the context of CW-complexes [19, Theorem 3.1]to the setting of posets. This result can be seen a combinatorial ana-logue of a theorem due to Conley [24, Theorem 1.2] [11]. Again, weassume that our coefficients are any principal ideal domain R . Fromnow on the poset X is assumed to be homologically admissible.Given a subposet Y ⊂ X we denote by ¯ Y the subposet ∪ x ∈ Y U x andby ˙ Y = ¯ Y − Y . Definition 4.2.1.
For each k ≥
0, we define m k = X basic sets Λ i rank H k ( ¯Λ i , ˙Λ i ) . Observe that in the particular case we have a Morse matching, thenthe basic sets are just critical points and m k is the number of criticalpoints of index k . Lemma 4.2.2.
If the index of the basic set Λ i is p , then H k ( ¯Λ i , ˙Λ i ) = 0 unless k = p, p + 1 . Moreover, if Λ i is just a critical point x ( p ) , then H k ( ¯Λ i , ˙Λ i ) = 0 for k = p and the principal domain of coefficients, R ,for k = p . ORSE-BOTT THEORY ON POSETS 13
Proof.
For convenience, during the proof we will denote Λ i = Λ. Sinceall the posets involved are cellular we can use cellular homology. Con-sider the Homology Long Exact Sequence for the pair ( ¯Λ , ˙Λ): · · · H p ( ¯Λ) H p ( ¯Λ , ˙Λ) H p − ( ˙Λ) H p − ( ¯Λ) H p − ( ¯Λ , ˙Λ) H p − ( ˙Λ) H p − ( ¯Λ) H p − ( ¯Λ , ˙Λ) H p − ( ˙Λ) · · · j ∂ ∼ = First of all, the homomorphism H k ( ˙Λ) → H k ( ¯Λ) is an isomorphism for k ≤ p −
2, so H k ( ¯Λ , ˙Λ) = 0 for k ≤ p −
2. Second, we have that: H p − ( ¯Λ , ˙Λ) = ker ∂ = Im j ∼ = H p − ( ¯Λ)ker j ∼ = H p − ( ¯Λ)Im i . Third, the homomorphism H p − ( ˙Λ) → H p − ( ¯Λ) induced by the in-clusion is surjective by the construction of cellular homology. There-fore H p − ( ¯Λ , ˙Λ) = 0. Fourth, if Λ is just a critical point x ( p ) , then H k ( ¯Λ , ˙Λ) = H k ( U x , b U x ) and by cellularity of X and the Homology LongExact Sequence for the pair ( U x , b U x ) the result follows. (cid:3) We denote by b k the Betti number of dimension k with coefficientsin the principal domain R .Taking into account the ideas involved in the proof of [19, Theorem3.1] and our Theorems 4.1.4 and 4.1.5 yields the Strong Morse-Bottinequalities: Theorem 4.2.3 (Strong Morse-Bott inequalities) . Let X be a homo-logically admissible poset and let M be a matching on X . Then, forevery k ≥ : m k − m k − + · · · + ( − k m ≥ b k − b k − + · · · + ( − k b . From the standard argument (see [32, p. 30]), we obtain the WeakMorse inequalities:
Corollary 4.2.4 (Weak Morse-Bott inequalities) . Let X be a homo-logically admissible poset and let M be a matching on X . Then: (1) For every k ≥ , m k ≥ b k . (2) χ ( X ) = P deg( X ) i =0 ( − k b k = P deg( X ) i =0 ( − k m k . Morse-Smale matchings.
In this section we generalise [18, Sec-tion 7] to the context of homologically admissible posets while improv-ing some of the results even in the case of simplicial or regular CW-complexes.Let X be a homologically admissible poset and let M be a Morse-Smale matching on X . We denote by c k the number os critical pointsof index k and by A k the number of prime closed M -orbits of index k . Denote by µ k the minimum number of generators of the torsionsubgroup T k of H k ( X ).Combining the proof of [18, Theorem 7.1] with our Pitcher strength-ening of Morse inequalities [16, Corollary 5.2.3] we obtain the followingimprovement of [18, Theorem 7.1], taking torsion into account. Theorem 4.3.1.
Let X be a homologically admissible poset and let M be a Morse-Smale matching on X . Let the coefficients R be a principalideal domain. Then, for every k ≥ : A k + k X i =0 ( − i c k − i ≥ µ k + k X i =0 ( − i b k − i . Definition 4.3.2.
Let X be a homologically admissible poset and let M be a Morse-Smale matching on X . Endow each element of X withan orientation. Let γ be an M -path γ : x ( p )0 ≺ y ( p +1)0 ≻ x ( p )1 ≺ y ( p +1)1 ≻ · · · ≺ y ( p +1) r − ≻ x ( p ) r . We define the multiplicity of γ by r − Y i =0 −h d ( p +1) y i , x i i p h d ( p +1) y i , x i +1 i p where d is the cellular boundary operator and h• , •i p is the inner prod-uct on C p ( X ) such that the degree p elements of X are mutually or-thogonal. Remark . Observe that the multiplicity of a path is always 1 or − Remark . The generalization of [19, Lemma 4.6] to our context isstraightforward.Both [19, Theorem 7.3] and [19, Corollary 7.4] generalise to our set-ting with the same proofs:
Theorem 4.3.3.
Let X be a homologically admissible poset and let M be a Morse-Smale matching on X . Let the coefficients be the field R . ORSE-BOTT THEORY ON POSETS 15
Denote by A ′ p the number of closed M -orbits of index p and multiplicity1. Then, for every k ≥ : A ′ k + k X i =0 ( − i c k − i ≥ k X i =0 ( − i b k − i ( R ) . Remark . While [19, Corollary 7.4] refined [19, Theorem 7.2], The-orem 4.3.3 does not refine our improved Theorem 4.3.1. They arecomplementary results.5.
Homological Lusternik-Schnirelmann Theorem
The purpose of this section is to prove a Lusternik-SchnirelmannTheorem for general matchings and a suitable definition of homologicalcategory.5.1.
Definition of the homological chain category and first prop-erties.
Let ( C ∗ , ∂ ) denote a free chain complex of abelian groups suchthat each term C p is finitely generated and only finitely many of the C p are non zero. We define the rank of C ∗ as rank ( C ∗ ) = P p rank ( C p ). Definition 5.1.1.
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 theinclusion i : B ∗ ֒ → C ∗ is a quasi-isomorphism. ) Let X be a topological space. For all the definitions that followwe consider coefficients in Z . We denote by S ∗ ( X ) its singular chaincomplex. Definition 5.1.2.
Let X be a topological space. We define its homo-logical chain category hccat( X ) = hccat( S ∗ ( X )).We introduce a homological lower bound for hccat( X ) analogous tothe Pitcher strengthening of Morse inequalities. Proposition 5.1.3.
Let X be a topological space with finitely generatedhomology. Then X k b k + 2 X k µ k ≤ hccat( X ) . Proof.
Let us denote by ( B ∗ , ∂ ) a bounded chain complex whose ho-mology is isomorphic to H ∗ ( X ). By standard algebra (see, for example[35, Theorem 4.11]), we have b k + µ k + µ k − ≤ rank ( B k ). Now theresult follows by a sum indexed by the dimension. (cid:3) Corollary 5.1.4.
Let X be a homologically admissible poset or a CW-complex with finitely generated homology. Then χ ( X ) ≤ hccat( X ) . In fact, the bound given by Proposition 5.1.3 is the best possible asa consequence of the following result due to Pitcher [34, Lemma 13.2].
Proposition 5.1.5.
Let ( C ∗ , ∂ ) be a free chain complex with singularhomology groups H k ( X ) , k = 0 , , . . . Denote by µ k the minimum num-ber of generators of the torsion subgroup T k of H k ( X ) and denote by b k the rank of H k ( X ) . Then there exists a free chain complex ( L, ∂ L ) such that: (1) For every k ≥ , the group L k has rank b k + µ k + µ k − . (2) There exists a monomorphism i : L ֒ → C which is a chain map. (3) The monomorphism i : L ֒ → C is a quasi-isomorphism. Corollary 5.1.6.
Let X be a topological space with finitely generatedhomology. Then hccat( X ) = X k b k + 2 X k µ k . Moreover, observe that a topological X is acyclic if and only ifhccat( X ) = 1.As a consequence of [12, Example 1.33] we have the following resultrelating the homological chain category to the Lusternik-Schnirelmanncategory: Proposition 5.1.7.
Let K be a simply connected CW-complex withfinitely generated homology groups such that there exists n satisfying H n ( K ) = 0 and H p ( K ) = 0 for p > n . Then cat( K ) ≤ hccat( K ) . The result does not necessarily hold if we remove the simply con-nectedness hypothesis, as the following example shows:
Example 5.1.8.
Consider the Poincar´e homology 3-sphere M . Ob-serve that hccat( M ) = hccat( S ) = 2. However, cat( M ) ≥ Homological Lusternik-Schnirelmann Theorem.
In this sub-section we state and prove a Lusternik-Schnirelmann Theorem for thehomological chain category and general matchings on posets.
Theorem 5.2.1.
Let X be a homologically admissible poset and let M be a Morse-Smale matching on X . Then hccat( X ) ≤ X basic sets Λ i hccat(Λ i ) . ORSE-BOTT THEORY ON POSETS 17
In particular, given Morse matching M on X , then hccat( X ) is a lowerbound for the number of critical elements of M .Proof. We will define a Morse matching M ∗ by means of perturbing M . The idea is to replace each prime closed orbit by two criticalpoints. This will be achieved by removing exactly one of the edges ofthe matching in each closed orbit. By repeating the technique usedin the proof of [19, Theorem 7.1], we obtain a Morse matching M ∗ satisfying m ∗ p = c p + A p + A p − , where m ∗ p denotes the number ofcritical points of index p of the matching M ∗ (see Subsection 4.3 forthe definition of A p ).Recall that C ∗ ( X ) denotes the cellular chain complex of X . Wedefine a map V : C p ( X ) → C p +1 ( X ) as follows: V ( x ) = ( − ǫ ( y, x ) y, if there exists y ∈ X with ( x, y ) ∈ M ∗ , , otherwise.Following the ideas of Minian [33], define the discrete flow operator φ : C p ( X ) → C p ( X ) as φ = id + dV + V d . The φ -invariant chains C φp ( X ) = { c ∈ C p ( X ) : φ ( c ) = c } form a well-defined subcomplex of ( C ∗ ( X ) , d ) [33]. Moreover, the inclu-sion of ( C φ ∗ ( X ) , d ) into ( C ∗ ( X ) , d ) induces isomorphisms in homologyand C φp ( X ) is isomorphic to the free abelian group spanned by thecritical p -elements of X [33]. As a consequence:(1) hccat( C ∗ ( X )) ≤ X p m ∗ p = X p c p + A p + A p − . There are two kinds of basic sets for M : critical points and disjointclosed M -orbits. Observe that if Λ i is a critical point, then hccat(Λ i ) =1 while if Λ i is a closed orbit, then hccat(Λ i ) = 2. So, from Equation(1), it follows that:hccat( C ∗ ( X )) ≤ X basic sets Λ i hccat(Λ i ) . Finally, observe that hccat( X ) = hccat( C ∗ ( X )) due to the isomorphismbetween cellular homology and singular homology for cellular posets(Theorem 2.3.3). (cid:3) Remark . In the proof of Theorem 5.2.1, Equation (1) could also bederived as a consequence of combining our Pitcher strengthening ofMorse-inequalities [16, Corollary 5.2.3] applied to the matching M ∗ with Corollary 5.1.6. As a consequence of [16, Theorem 3.3.6], we obtain the followingcorollary:
Corollary 5.2.2.
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Geometriccombinatorics , pages 497–615. Providence, RI: American Math-ematical Society (AMS); Princeton, NJ: Institute for AdvancedStudies, 2007. ISBN 978-0-8218-3736-8/hbk. (1)
Departamento de Geometr´ıa y Topolog´ıa, Universidad de Sevilla,Spain., (2)
Instituto de Matemticas, Universidade de Santiago de Com-postela, Spain.
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