Fundamental theorems of Morse theory on posets

http://www.aimspress.com/journal/Math AIMS Mathematics, 7(8): 14922–14945. DOI: 10.3934/math.2022818 Received: 27 March 2022 Revised: 24 May 2022 Accepted: 26 May 2022 Published: 13 June 2022 Research article Fundamental theorems of Morse theory on posets D. Fern´ andez-Ternero1, E. Mac´ ıas-Virg´ os2,*, D. Mosquera-Lois2, N. A. Scoville3and J. A. Vilches1 1Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Sevilla, Spain 2CITMAga, Universidade de Santiago de Compostela, Spain 3Department of Mathematics and Computer Science, Ursinus College, Collegeville PA 19426 U.S.A. *Correspondence: Email: [email protected]; Tel: +34881813153; Fax: +34881813197. Abstract: We prove a version of the fundamental theorems of Morse theory in the setting of finite partially ordered sets. By using these results we extend Forman’s discrete Morse theory to more general cell complexes and derive the Morse-Pitcher inequalities in that context. Keywords: Morse theory; finite spaces; posets; two-wide posets; down-wide posets Mathematics Subject Classification: 06A07, 37B35, 55U99 1. Introduction Morse theory was originally introduced as a tool for the study of variational problems on manifolds by relating the homology of a space with some critical objects arising after defining a map and considering its induced dynamics ([25]). Since its introduction, Morse theory has been an active field of research and connections with many different areas of mathematics have been found. That interaction has led to several adaptations of Morse theory in different contexts: PL versions by Banchoff([2, 3]), by Bestvina and Brady ([8]), and a purely combinatorial approach by Forman ([15, 17]). Nowadays, not only pure mathematics benefit from that interaction but also applied mathematics ([18]) due to the importance of discrete settings. Roughly speaking, Morse theory addresses the study of the topology (homotopy or 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 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 ([15, 23]). Finite spaces and finite partially ordered sets (posets) are essentially the same objects considered from different points of view, as observed by Alexandroff([1]). In this way, Stong ([29]) proved how 14923 the combinatorics of posets classify the (strong) homotopy types of their associated finite spaces. These ideas were continued in the work of Raptis ([28]) on the homotopy theory of posets. More or less at the same time as Stong’s work, McCord ([22]) discovered the relationship between finite spaces and finite simplicial complexes. Given a finite space X, its associated “order complex” K(X) is a simplicial complex having the same weak homotopy type as X. Conversely, for each finite simplicial complex K, its “face poset” X(K) is a finite space which is weak homotopy equivalent to K. In this way, weak homotopy types of finite spaces coincide with homotopy types of finite simplicial or regular CW-complexes. Notice that by Whitehead’s Theorem, the distinction between weak homotopy types and homotopy types is lost when we only consider the associated complexes, while for finite spaces there is a distinction between weak homotopy types and homotopy types. The Morse theory developed in this work sees at most the simple homotopy type of the posets but does not recover the (strong) homotopy type (see Example 4.6). This phenomenon may suggest that it is possible to study the Morse theory of posets by studying a Morse theory in their associated order complexes. However, this is not possible since the order complex functor does not carry Morse functions nor matchings. Finite homology manifolds (see Subsection 2.2) are of special relevance among finite spaces. This is due to several reasons. First, these spaces play the role of simplicial homology manifolds in the setting of posets. Second, and related to the first one, studying finite homology manifolds provides information about the topology of simplicial homology manifolds. This is the reason why we develop our results for down-wide and two-wide posets, which are families of finite spaces that include the finite homology manifolds. The idea of approaching problems regarding posets by using topological methods dates back at least to the works of Brown ([10]), Quillen ([27]), and Stong ([30]) dealing with the way the homotopy type of the poset of non-trivial p-subgroups of a finite group Gordered by inclusion determines algebraic properties of the group. Recently, Barmak and Minian ([6]), and later Chocano, Mor´ on, and Ruiz del Portal ([12]) have made use of the topology of finite spaces in order to study groups as automorphism groups of certain posets. A general reference for different problems dealt with through the study of poset topology is the survey of Wachs ([31]). Therefore, it makes sense to study the (strong and weak) homotopy type and homology of finite spaces by means of some adapted version of Morse theory. The aim of this work is to develop an extension of Morse theory to finite posets introduced by Minian ([24]) in order to prove the Fundamental Theorems of Morse theory in this setting. More precisely, we study the evolution of level subposets with or without critical values. Moreover, some of the consequences of the Fundamental Theorems are exploited. For instance, we provide an alternative proof of Forman’s decomposition theorem [15, Corollary 3.5], an extension of discrete Morse theory to more general cell complexes, and the Morse-Pitcher inequalities. In order to achieve our goal, we work with finite topological spaces (that is, posets endowed with the Alexandrofftopology) with their intrinsic topology instead of using the topology of the associated (order) simplicial complex. This is for several reasons. First, it allows us to establish more general hypotheses and to take advantage of more efficient algorithms and computations. For the second and more crucial reason, we need to elaborate a bit. Given a simplicial complex Kand a Morse function f:K→R, the function induces a combinatorial Morse function on its face poset X(f): X(K)= X→Rsuch that X(f) retains the topological information regarding the level subcomplexes of K (see Section 5). However, this technique does not work well the other way around. That is, given a AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14924 poset Xand a discrete Morse function f:X→R, there is no canonical way of constructing a discrete Morse function on its order complex K(X) such that it retains the topological information of the level subposets of X. Therefore, it is more general to study Morse theory in the setting of posets than in the context of simplicial or regular cell complexes, and the first setting can not be reduced to the second one. The organization of this paper is as follows. In Section 2 we recall some definitions and standard results about posets and finite topological spaces originally stated by Alexandroff([1]) and Stong ([29]). Section 3 is devoted to the study of discrete Morse theory in the context of posets, proving the correspondence between Morse matchings and order preserving Morse functions on two-wide graded posets. In Section 4 we prove the Fundamental Theorems of Morse theory in this setting. Among these results are the so-called Structural Theorems, which show that the topology of the level subsets does not change in absence of critical values and how it changes when some critical point arises. We then show an example that illustrates the results of this section. Finally in Section 5, we show some consequences of these results, such as extending Forman’s original result ([15, Corollary 3.5]) regarding the homotopy type of a regular CW-complex with a Morse function defined on it and obtaining the Morse inequalities. Moreover, we study a method to reduce the number of critical points of a Morse function defined on a poset (Theorem 5.7). 2. Finite spaces, posets, and simplicial complexes This section is devoted to introducing the objects we will work with. In particular, we are interested in two kinds of posets: two-wide (introduced by Bloch ([9])), and down-wide. It is in this setting where we shall establish our main results. Most of the material presented in this section is well established in the literature. For further details, the reader is referred to [4, 7, 9, 14, 24, 31]. All posets will be assumed to be finite. By a finite space, we mean a finite T0-space. 2.1. Preliminaries on finite posets It is well known that finite posets and finite T0-spaces are in bijective correspondence ([1]). If (X,≤) is a poset, a topology on Xis defined by taking as basis the open sets Ux:={y∈X:y≤x} for each x∈X. On the other hand, if Xis a finite T0-space, for any x∈X, define the minimal open set Uxas the intersection of all open sets containing x. Then Xmay be given a poset structure by defining y≤xif and only if Uy⊆Ux. It is easy to see that these correspondences are mutually 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. As a consequence of the correspondence between posets and finite T0-spaces, we will use both notions interchangeably. We need to introduce some basic notions and results. Definition 2.1. A chain in the poset X is a subset C ⊆X such that if x,y∈C, then either x ≤y or y≤x. The chain x0<x1< . . . < xnhas length n. The height of X is the maximum length of the chains in X. The height of an element x ∈X is the height of Uxwith the induced order. 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 Uxis AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14925 homogeneous for every x ∈X. In that case, the degree of x is its height. If x ∈X has height p, then we denote x =x(p). We denote by deg(X)the degree of any maximal element. Let Xbe a finite poset, x,y∈X. If x<yand there is no z∈Xsuch that x<z<y, we write x≺y. For x∈Xwe also define b Ux:={w∈X:w<x}as well as Fx:={y∈X:y≥x}and b Fx:={y∈X:y>x}. We denote b Cx=(Ux∪Fx)− {x}. Due to the correspondence between posets and finite topological spaces we can also study the homotopy type of a poset by means of this intrinsic topology. The homotopy type of finite spaces can be studied combinatorially by using the notion of beat points. This was first developed by Stong ([29]). Definition 2.2. ([29]) A point x ∈X is a beat point if either b Uxhas a maximum (down beat point) or b Fxhas a minimum (up beat point). Notice that, by Corollary 3 in [29], a poset with a maximum or a minimum is contractible. Consequently, if x∈Xis a down beat point (respectively up beat point) then b Uxis contractible (respectively b Fxis contractible). The next proposition states that removing beat points from a poset does not change its homotopy type as a T0-space. Proposition 2.3. [4, Proposition 1.3.4] Let x ∈X be a beat point. Then X −{x}is a strong deformation retract of X. We now recall McCord functors between posets and simplicial complexes. The reader less familiarized with this topic may consult [4]. Given a poset X, we define its order complex K(X) as the simplicial complex whose k-simplices are the non-empty chains of Xof length k. Furthermore, given an order preserving map f:X→Ybetween posets, we define a simplicial map K(f): K(X)→ K(Y) by K(f)(x)=f(x). Conversely, if Kis a simplicial complex, we define the face poset of K, denoted by X(K), as the poset of simplices of Kordered by inclusion. Given a simplicial map φ:K→Lwe define the order preserving map X(φ): X(K)→ X(L) by X(φ)(σ)=φ(σ) for each simplex σof K. The face poset functor is defined analogously for CW-complexes. That is, given a CW-complex K, X(K) is the poset of cells of Kordered by inclusion. Given a cellular map φ:K→Lwe define the order preserving map X(φ): X(K)→ X(L) by X(φ)(σ)=φ(σ) for each cell σof K. Note that for a simplicial complex K,KX(K) is sd(K), the first barycentric subdivision of K. Recall that |K|denotes the geometric realization of a complex K. We will now recall two results which relate the topology of posets and that of their order complexes and the topology of complexes and that of their face posets. These results will be very useful in order to obtain information about the homotopy type of complexes equipped with a discrete Morse function without using simple homotopy type. For details and a proof of the following result, see [4, p.12–15]: Theorem 2.4. The following statements hold: 1. Let X be a finite T0-space. Then there is a map µX:|K(X)| → X which is a weak homotopy equivalence. 2. Let K be a simplicial complex. Then there is a map µK:|K|→X(K)which is a weak homotopy equivalence. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14926 We recall now McCord’s Theorem, which plays a central role in the proof of Theorem 2.4. Let Xbe a topological space. An open cover Uof Xis called a basis like open cover for Xif Uis a basis for a topology in the underlying set of X. Note that given a finite space X, the cover by minimal open sets {Ux}x∈Xis a basis like open cover. We reproduce the statement of McCord’s Theorem: Theorem 2.5. [22, Theorem 6] Let f :X→Y be a continuous map between topological spaces. Suppose there is a basis like open cover Uof Y such that for every U ∈ U, the restriction f|f−1(U):f−1(U)→U is a weak homotopy equivalence. Then f :X→Y is a weak homotopy equivalence. 2.2. Two-wide and down-wide posets In this subsection, we work with two classes of posets which will play a key role in the later development of Morse theory in the context of finite posets. Definition 2.6 ([9]).A poset X is two-wide if for any x,z,y∈X such that x ≺z≺y, there is some z0∈X such that z0,z and x ≺z0≺y. Remark 2.7. Let X be a finite poset. If X is two-wide, then for any pair of elements x,y∈X such that x<y and x ⊀y, #{z:x≤z≤y} ≥ 4. It is easy to show that face posets of regular CW-complexes are two-wide. We recall this result below. Lemma 2.8. [21, Lemma 4.1, p. 168] Given a regular CW-complex K, its face poset X(K)is two-wide. We now introduce the notion of down-wide poset, which is one of the properties that satisfy the family of posets we are interested in. Definition 2.9. Given a poset X and x ∈X, we define the down-incidence number of x as the cardinality of the set ∂x={y∈X:y≺x}. The poset X is down-wide if #∂x,1for every x in X. Observe that a down-wide poset is exactly a poset without down beat points, and ∂xis the set of maximal elements of ˆ Ux. It is easy to check the following result: Lemma 2.10. For any regular CW-complex K, its face poset X(K)is down-wide. Therefore all posets coming from simplicial complexes by the McCord functor Xare down-wide. However, not every down-wide poset is the face poset of some simplicial complex. Example 2.11. The poset X pictured in Figure 1 is down-wide. However, it is not the face poset of any simplicial complex K. Otherwise K would have two 0-simplices and two 1-simplices. Figure 1. A down-wide poset which is not the face poset of any simplicial complex. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14927 Definition 2.12 ([24]).A poset X is a model of a CW-complex K if the geometric realization of K(X) is homotopy equivalent to K. Observe that a subposet Yof a poset Xis an open subspace with the finite topology if whenever x∈Yand y≤x, then y∈Y. Therefore, an open subspace of a down-wide poset is down-wide, and an open subspace of a two-wide poset is two-wide. Note that the properties of being two-wide and down-wide do not imply each other. For example, the poset depicted in Figure 3 is down-wide but it is not two-wide. Conversely, the poset depicted in Figure 2 is two-wide but it is not down-wide. Figure 2. A two-wide poset which is not down-wide. 5 5 5 22 3 3 44 6 w y Figure 3. Lemma 3.6 may not hold for posets which are not two-wide. We recall the notion of homologically admissible edges introduced by Minian ([24]). We denote by H(X) the Hasse diagram associated to the poset X; that is, the directed graph with edges (x,y) for each x≺yin the poset X. Definition 2.13 ([24]).Let X be a poset. An edge (w,x)∈ H(X)is homologically admissible if b Ux−{w} is acyclic. A poset is homologically admissible if all its edges are homologically admissible. Definition 2.14. Let X be a poset. An edge (w,x)∈ H(X)is 1-admissible if b Ux− {w}is simply connected. A poset is 1-admissible if all its edges are 1-admissible. Remark 2.15. The face poset of a regular CW-complex is homologically admissible [24, Remark 2.13]. A simplicial complex Kis a closed homology manifold of dimension nif the link of every k-simplex has the homology of a sphere Sn−k−1. A poset Xis a finite closed homology manifold if its order complex K(X) is a closed homology manifold. By [24, Theorem 4.6], if the poset Xis a finite closed homology manifold, then it is homologically admissible. As a consequence, the theory that we will develop for posets which are homologically admissible can be applied to study the topology of closed triangulable homology manifolds by means of their triangulations. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14928 We end the subsection by relating the properties of being homologically admissible with those of being two-wide and down-wide. This is important since it guarantees that all results in this work hold for homologically admissible posets. In order to prove that homologically admissible posets are down-wide and two-wide, we recall a particular case of a construction of cellular homology for posets due to Farmer ([14]) and Minian ([24]). Given a homologically admissible poset X, we define its cellular homology with Z2coefficients. First, we define its cellular chain complex (C∗,d) as follows: for each p,Cp(X) is a Z2-vector space (where Z2stands for the field with two elements) with one generator for each element of Xof degree p. The differential d:Cp(X)→Cp−1(X) is defined as d(x)=Pw≺xw. It follows that H∗(C∗(X))  H∗(X;Z2) ([24, Theorem 3.7]) where H∗(X;Z2) stands for the singular homology with Z2coefficients. Lemma 2.16. Let X be a poset. If X is homologically admissible, then it is down-wide. Proof. For any x(1) and w≺x, since b Ux− {w}is acyclic, then b Ux− {w}is non-empty, so there exists w0,wsuch that w0≺x. For any x(p)with p>1, suppose that there is a unique w∈Xsuch that w≺x. We will arrive to a contradiction. Using cellular homology, d2(x)=d(w)=X q≺w q,0 which is a contradiction.  Proposition 2.17. Let X be a poset. If X is homologically admissible, then it is two-wide. Proof. Suppose there are elements x≺z≺y. We have to show that there is some z0,zsuch that x≺z0≺y. Using cellular homology, dy =±z+X ˜z,z,˜z≺y ±˜z and dz =±x+X ˜x,x,˜x≺z ±˜x. Since d2=0, 0=d2y=±dz +X ˜z,z,˜z≺y ±d˜z=±x+X ˜x,x,˜x≺z ±˜x+X ˜z,z,˜z≺y ±d˜z. Since this equation holds, there must be some z0,zsuch that dz0=±x+X ˜x,x,˜x≺z0 ±˜x, that is, there is some z0,zsuch that x≺z0≺y. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14929 3. Morse Theory on posets 3.1. Definition of Morse functions We recall the definition of Morse functions for posets introduced by Minian ([24]). It is an adaptation of Forman’s theory ([15, 17]) to the context of posets. Definition 3.1. Let X be a finite poset. A Morse function on X is a function f :X→Rsuch that, for every x ∈X, we have #{y∈X:x≺y and f (x)≥f(y)} ≤ 1 and #{w∈X:w≺x and f (w)≥f(x)} ≤ 1. Definition 3.2. If f is a Morse function on a finite poset X, the point x ∈X is said to be critical if #{y∈X:x≺y and f (x)≥f(y)}=0 and #{w∈X:w≺x and f (w)≥f(x)}=0. The set of critical points is denoted by crit f. The images of the critical points are called critical values. The points (respectively values) which are not critical are said to be regular points (respectively regular values). 3.2. Technical Lemmas for Morse functions We begin by stating a result that plays the role of two important theorems developed by Forman in the simplicial setting ([15, Theorems 1.2 and 1.3]). In fact, the proof of the following Key Lemma is much simpler in this context than in the simplicial setting. Lemma 3.3 (Key Lemma).Suppose that X is a finite two-wide poset and there are two elements w <y such that w ⊀y. Then there are elements x ,˜x such that w ≺x<y and w ≺˜x<y. Proof. Since w<yand w⊀y, there is a chain w≺x0≺ · · · ≺ xn≺ywith n≥0. If n=0, the result just follows from the definition. Otherwise, apply the definition to the chain w≺x0≺x1. Therefore, there is an ˜x0,x0verifying that w≺x0<x1and w≺˜x0<x1. Take x=x0and ˜x=˜x0. It holds that w≺x<yand w≺˜x<yas we wanted to prove.  Remark 3.4. Observe that Lemma 3.3 does not hold in general for finite posets. As an example, consider the poset of Figure 3, and take the points labelled as w and y. Definition 3.5. Given a poset X, a Morse function f :X→Ris said to satisfy the Exclusion condition if for every regular point x ∈X, exactly one of the following conditions holds: 1. There exists exactly one y ∈X, x ≺y, such that f (x)≥f(y). 2. There exists exactly one w ∈X, w ≺x, such that f (w)≥f(x). The following result plays the role of the Exclusion Lemma ([15, Lemma 2.5]) in the context of finite spaces. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14930 Lemma 3.6 (Exclusion Lemma).Let X be a finite two-wide poset and f :X→Ra Morse function on X. Then f :X→Rsatisfies the Exclusion condition. Proof. Let xbe a regular point. Since xis not critical, then at least one of the conditions in Definition 3.5 holds. We will see that these conditions are mutually exclusive. By way of contradiction, suppose that both conditions hold, and take x≺yand w≺xas in conditions 1 and 2, respectively. Then w≺x≺y. Since Xis two-wide, by the Key Lemma (Lemma 3.3) there exists x0,xsuch that w≺x0≺y. By the definition of Morse function applied to wwith w≺x0, we get f(w)<f(x0) since we already have f(w)≥f(x) and w≺x. Similarly with yand x0≺y, we obtain f(x0)<f(y)≤f(x). As a consequence we obtain the following chain of inequalities: f(x)≤f(w)<f(x0)<f(y)≤f(x). Therefore f(x)<f(x), which is a contradiction.  It is interesting to point out that the Exclusion Lemma does not necessarily hold in general for posets which are not two-wide, as the following example shows. Example 3.7. Lemma 3.6 may not hold for arbitrary posets as we show below. Consider the downwide model of S3(taken from [24, Fig.2]) with the Morse function f represented in Figure 3 by the labelling of the points. 3.3. Matchings We recall the definition of a (Morse) matching introduced by Chari ([11]) and further developed by Minian ([24]). Definition 3.8. A matching Min a poset X is a subset M ⊆ X×X such that •(x,y)∈ M implies x ≺y; •each x ∈X is in at most one element in M. We introduce some terminology which will be used later. Given a matching Mon the poset X, we will decompose Xas the disjoint union of three subsets X=crit(M)ts(M)tt(M). For each edge (x,y)∈ M, we say that xis the source of the edge and yis the target. We define the source of the matching s(M) as the set whose elements are the sources of the edges in the matching. Analogously, we define the target of the matching t(M) as the set whose elements are the targets of the edges in the matching. 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 3.9. A matching is homologically admissible if each element of the matching is homologically admissible. The notion of a 1-admissible matching is defined analogously. Let H(X) be the Hasse diagram of a poset X. If Mis a matching in X, write HM(X) for the directed graph obtained from H(X) by reversing the orientations of the edges which are not in M. Definition 3.10. The matching Mis a Morse matching on X if HM(X)is acyclic as a directed graph. Any point of H(X)not incident with an edge of Mis called critical. The set of all critical points of M is denoted by crit(M). AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14937 f−1((ai,bi]) ∩ {vj}j={vi}and Xf0 a=Xf aand Xf0 b=Xf b. So, we can assume that there is only one homologically admissible edge (w,v) for the Morse matching associated to the function fin (a,b]. Second, since wis an up beat point in Xb,i:Xb− {w},→Xbis a homotopy equivalence, so it is enough to prove that i:Xa=Xb− {w,v},→Xb− {w}induces an isomorphism in homology. By applying the Long Exact Sequence of homology to the pair (Xb− {w},Xa), it follows that i:Xa,→Xb− {w}induces an isomorphism in all homology groups if and only if H∗(Xb− {w},Xa)0. As a consequence of the Excision Theorem [20, Theorem 2.20], given two open sets Aand Bwhich cover Xb− {w}, then there is an isomorphism H∗(B,A∩B)H∗(Xb− {w},A). Considering A=Xaand B=Uv, it follows that H∗(Uv,b Uv− {w})H∗(Xb− {w},Xa). Since w≺vis an element in the matching and the matching is homologically admissible, then b Uv−{w} is acyclic. By applying the Long Exact Sequence of homology to the pair (Uv,b Uv− {w}) and using the fact that Uvis contractible, it follows that H∗(Uv,b Uv− {w})H∗(b Uv− {w}), so H∗(Uv,b Uv− {w})0 if and only if the element of the matching w≺vis homologically admissible.  Remark 4.9. In the above proof, b Uv− {w}is always nonempty due to the fact that X is a down-wide poset. Theorem 4.10. Let X be a finite, path-connected, down-wide, two-wide poset. Let f :X→Rbe a discrete Morse function. Suppose that (a,b]for a <b contains no critical values of f . If the Morse matching associated to the function f is homologically admissible, then the inclusion i:Xa,→Xb induces an isomorphism in homology. Proof. This follows by combining Propositions 2.3, 4.7, and 4.8.  Our next goal is to state the weak homotopical collapsing theorem, where again it is necessary to use the topology of the posets rather than their combinatorial properties. We need to add the extra hypothesis that the Morse matching associated to the function fis 1-admissible. There is a weaker notion of beat point which we recall now. Definition 4.11. ([5],[4, Definition 6.2.1]) The point x ∈X is a γ−point if b Cx=(Ux∪Fx)− {x}is homotopically trivial. Proposition 4.12. Under the conditions of Proposition 4.7, the inclusion i: (Xb−{w})−{v},→Xb−{w} is a weak homotopy equivalence if the element w ≺v of the Morse matching associated to the function f is 1-admissible and homologically admissible. Moreover, v is a γ−point in Xb− {w}. Proof. We will apply McCord’s Theorem (Theorem 2.5) to the base {Ux:x∈Xb− {w}}. There are two cases to consider: 1. If x,v, then i−1(Ux) has a maximum and therefore is contractible, so i|i−1(Ux):i−1(Ux)→Uxis a weak homotopy equivalence. 2. If x=v, then i|i−1(Ux):i−1(Ux)→Uxis the map i:b Uv−{w},→Uv. The subspace Uvis contractible so it is homotopically trivial. Therefore i:b Uv− {w},→Uvis a weak homotopy equivalence if and only if b Uv− {w}is homotopically trivial. Now, since b Uv− {w}is simply connected and acyclic, by Hurewicz Theorem it is homotopically trivial.  AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14938 Theorem 4.13. Let X be a finite path-connected down-wide and two-wide poset. Let f :X→Rbe a discrete Morse function. Assume that (a,b]for a <b contains no critical values of f . 1. If the Morse matching associated to the function f is 1-admissible and homologically admissible, then the inclusion i:Xa,→Xbis a weak homotopy equivalence. 2. Moreover, in case (a,b]contains no critical values of f and contains at most one regular value f(v). Then Xb=Xaor Xb−Xa={v,w}where w ≺v and w is an up beat point in Xband v is a γ−point in Xb− {w}. Proof. The result follows by combining Propositions 2.3, 4.7, and 4.12.  Remark 4.14. Theorem 4.13 does not necessarily hold for arbitrary posets, as Example 4.5 shows. The following result explains what happens with the homotopy type when we reach critical values. It plays the role of [23, Theorem 3.2] in the case of smooth Morse theory and [15, Theorem 3.4] in discrete Morse theory. Recall from Definition 2.9 that for a poset Xand x∈X,∂x={y∈X:y≺x}. Theorem 4.15. Let X be a path-connected finite poset and f :X→Ran order preserving Morse function. Suppose that f −1(a,b]={x}is a critical element. Then Xb=Xa∪∂x(p)x(p), that is, Xbis obtained from the poset Xaby adding a new element x above the elements of the subposet ∂x⊆Xa. Proof. If y>x, then f(x)≤f(y) by the order preserving hypothesis. Moreover, since f−1(a,b]={x} for a<b,a<f(x)<f(y) and f(y)>b. Furthermore, if y∈Xa, then y≤zfor some zwith f(z)≤a. Hence b<f(y)≤f(z)≤a, a contradiction. Therefore Xa∩Fx=∅. If y<x, an analogous reasoning shows that f(y)≤a. Therefore, Xb=Xa∪Uxand Xa∩Ux=ˆ Ux. In combination with the result of the previous paragraph, we arrive at the desired conclusion.  Remark 4.16. Theorem 4.15 is easily generalized to the case where f −1(a,b]is a collection of critical elements. The following example illustrates Theorems 4.10 and 4.15 by working out, step by step, the changes in homology of the level subposets of a poset endowed with a Morse function. Example 4.17. Let us denote by X the finite model of RP2depicted in Figure 7 (see [4, Example 7.1.1], [19, Proposition 4.1] and [13, p. 138]). It can be checked that it is two-wide, down-wide, and homologically admissible. Consider the function f :X→Rgiven by the values depicted at the right side of the elements of X. It is clear that f is a Morse function. We will denote the level subposets by Xt. 0 5 1 2 3 4 12 9 8 11 10 6 7 x1 w1w2w3 x2x3x4x5x6 y1y2y3y4 Figure 7. Morse function on a finite model of RP2. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14939 We begin the analysis of the level subposets. First, as Proposition 4.1 claims, the minimum value of f , corresponding to the element w1, is a critical value (we are adding a path-connected component) (see Figure 8): 0 w1 Figure 8. Level subposet RP2 0. As we reach the value t =1, the inclusion i:X0→X1induces an isomorphism in homology. Observe that X1is contractible by removing beat points (see Figure 9 (a)). The situation does not change when we reach the value t =3since X3is still contractible by removing beat points (see Figure 9 (b)). The value t =5is critical and the map i:X3→X5does not longer induce an isomorphism in homology. Observe that X5has the homotopy type, by removing beat points, of a finite model of the circle S1(see Figure 9 (c)). 0 1 2 w1w2 x2 0 1 2 3 4 w1w2w3 x2x4 0 5 1 2 3 4 w1w2w3 x2x3x4 (a) (b) (c) Figure 9. Level subposets RP2 1,RP2 3and RP2 5. The value t =6is regular and it can be checked that the map i:X5→X6induces an isomorphism in homology (see Figure 10 (a)). 0 5 1 2 3 4 6 7 w1w2w3 x2x3x4x5 y3 0 5 1 2 3 4 9 8 6 7 w1w2w3 x2x3x4x5x6 y3y4 (a) (b) Figure 10. Level subposets RP2 6and RP2 8. The situation does not change when we reach the value t =8(see Figure 10 (b)) nor the value t=10 (see Figure 11). 0 5 1 2 3 4 9 8 11 10 6 7 x1 w1w2w3 x2x3x4x5x6 y1y3y4 Figure 11. Level subposet RP2 10. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14940 Finally, we reach the value t =12, which is critical. The map i:X9→X12 =X induces an isomorphism in homology despite t =12 is a critical value. 5. Consequences 5.1. Forman’s Theory without simple homotopy types It is clear that for any regular CW-complex K, given a Morse function f:K→R, it induces a combinatorial Morse function on its face poset X(f): X(K)=X→Rsuch that the face poset functor satisfies X(Ka)=X(K)a, where Kadenotes the level subcomplex corresponding to the value a, that is, the subcomplex of Kdefined as follows Ka=[ f(τ)>a [ σ≤τ σ. It is worthwhile to mention that a function f:K→Rdefined on the collection of simplices or regular cells of Kis a (discrete) Morse function if and only if X(f): X(K)=X→Ris a Morse function on the corresponding poset. Therefore, as a first consequence, by using Theorem 2.4 (2), we recover Forman’s result [15, Corollary 3.5] without the use of simple homotopy types. 5.2. Morse-Pitcher inequalities Another consequence of our structural theorems of Morse Theory for finite spaces is that we can reproduce step by step the classical proof (see [26, 23] for the standard argument) of Morse inequalities in this new context of posets. We consider coefficients in a principal ideal domain. Definition 5.1. Let f :X→Rbe a Morse function. We denote by mithe number of critical points of height i and by bithe Betti number of dimension i. Corollary 5.2 (Strong Morse inequalities).Let X be a down-wide and two wide poset and let f :X→R be an order preserving Morse function. Suppose that the Morse matching associated to f is homologically admissible and homology-regular. Then for every i ≥0and domain of coefficients: mi−mi−1+· · · +(−1)im0≥bi−bi−1+· · · +(−1)ib0. Corollary 5.3 (Weak Morse inequalities).Let X be a down-wide and two wide poset and let f :X→R be an order preserving Morse function. Suppose that the Morse matching associated to f is homologically admissible and homology-regular. Then 1. mi≥bifor every i. 2. The Euler-Poincar´e Characteristic satisfies χ(X)= deg(X) X i=0 (−1)ibi= deg(X) X i=0 (−1)imi. Remark 5.4. The Morse inequalities for homologically admissible posets can also be derived by following a combinatorial Hodge-theoretic argument mimicking [16] since the arguments provided by Forman can be reproduced without changes in this context. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14941 Our next goal is to strengthen the Morse inequalities in the spirit of Pitcher’s approach for smooth Morse theory ([26]). In order to proceed, we introduce some notation and an auxiliary result from homological algebra. Let as denote by {c0,...,ck,...,cn}the image of the Morse function f:X→R. Then, there exist real numbers {ak}ksatisfying: c0<a0<c1<a1· · · <ck<ak<···cn<an. To simplify our notation, we will denote Xaiby Xi. We denote the coefficient ring, which is assumed to be a principal ideal domain, by R. A a consequence of the Structure Theorem for finitely generated modules over a principal ideal domain, it follows that: Hi(X)Rbi⊕R (r1)⊕ · · · ⊕ R (rηi). For the case of relative homology, we use the notation: Hi(Xk,Xk−1)Rbk i⊕R (r1)⊕ · · · ⊕ R (rηk i). Set Mk i=bk i+ηk i+ηk−1 iand Mi=Pk=0Mk i. Proposition 5.5 ([26, Theorem 14.1]).Let (C∗(X), ∂)be a free chain complex with singular homology groups Hi(X), i =0,1, . . .. Then there exists a free chain complex (L, ∂L)such that: 1. For every i ≥0, the group Lihas rank Mi. 2. There exists a monomorphism i:L,→C which is a chain map an induces isomorphims in homology. Theorem 5.6 (Pitcher strengthening of Morse inequalities).Let X be a down-wide and two wide poset and let f :X→Rbe an order preserving Morse function whose associated Morse matching is homologically admissible and homology-regular. Then 1. For every i ≥0: mi≥bi+ηi+ηi−1. 2. For every i ≥0: mi−mi−1+· · · +(−1)im0≥bi−bi−1+· · · +(−1)ib0+ηi. Moreover, the equality is attained for i =deg(X). Proof. First, apply Theorem 5.5 to the singular chain complex of X. Now, observe that: rank Li=rank Hi(X)+rank Im ∂L i+1+rank Im ∂L i. The first set of inequalities follows from observing that rank Im ∂L i≥ηi−1. The second set of inequalities follows from taking alternating sums in i. Observe that if deg(X)=n, then µn=0 since Hn(X) is a submodule of the free module Cn(X). Moreover, µ0=0 and µ−1is defined as 0. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14942 5.3. Cancelling critical points Both the Morse and Morse-Pitcher inequalities suggest the study of Morse functions with few critical points, the so-called optimal Morse functions. In order to obtain such functions we present an approach consisting of canceling pairs of critical elements, extending to the context of posets known results on smooth manifolds and simplicial complexes. We present a result which can be seen as the adaptation of [15, Theorem 11.1] to our context. Theorem 5.7 (Cancelling critical points).Given a Morse matching Mon a finite graded poset X, assume that z(p+1) and x(p)are critical points such that there is a unique M-path γ:zy=x0≺z0x1≺z1 · · · ≺ zrxr=x with y(p)≺z(p+1) (there is no other M-path from any p-face of z(p+1) to x(p)). Then there is a matching M0such that: •The set of critical points of M0is crit(M0)=crit(M)− {x,z}. •Moreover, M0=Mexcept along the unique M-path from ∂z to x. Proof. We define M0as follows: 1. tM0(w)=tM(w) if w<{y,z0,x1,z1, . . . zr,x}(M0=Mexcept along the unique gradient path from ∂zto x) 2. tM0(xi)=zi−1,i=1,...,r(we reverse the gradient path from xto z0so xis no longer critical) 3. tM0(y)=z(we reverse the arrow from yto zso zis no longer critical). It remains to check that there are no closed M0-paths. We argue by contradiction. Suppose there was a closed M0-path δ. Claim. Under the above hypothesis, δwould contain at least one p-element from γand one p-element not in γ. Proof of the Claim. The elements coming from γcan not give a closed M0-path on their own since we have just reverted their arrows. The elements of Xwhich are not in γcan not give a closed M0-path since in that case we would also have a closed M-path and Mis a gradient vector field. Therefore in δ we must have at least one p-element in each of their sets.  Hence, δwould contain a sequence of the form: xi≺w0s1≺w1 · · · ≺ wsxj where s≥0, wl,xk,wl,zk,sl,xk,sl,zk, for all land k. Since tM0(wl)=tM(wl) and tM0(sl)=tM(sl) for all l, we have a M-path: w0s1≺w1 · · · ≺ wsxj. Let us consider two cases: AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14943 1. If i,0, then s1,xi−1,xiand s1≺tM0(xi)=zi−1. Therefore, we can define a second gradient M-path γ0,γfrom ∂zto x: γ0:y=x0≺z0x1≺ · · ·  xi−1≺zi−1s1≺w1 · · · xj≺zj · · ·  xr=x. Which is a contradiction. 2. If i=0, then y=x0,s1≺tM0(y)=z. Therefore, we can define the following M-path: γ0:zs1≺w1 · · · ≺ wsxj≺zj · · ·  xr=x which is different from γand also goes from ∂zto x. Then we have a contradiction.  Finally, there is a kind of dual result to Theorem 5.7 which allows us to create critical points. Both the statement and the proof are a straightforward translation of [15, Theorem 11.3]. 6. Conclusions We have studied Morse theory in the setting of finite partially ordered sets. Our approach covered Formans's discrete Morse theory for cell complexes as a particular case. Contrary to what happens for cell complexes, the Morse theory we studied in the setting of posets does not allow to control homotopy type. Still, we have proved two collapsing theorems and an adjunction theorem. The first collapsing result is a homological collapsing theorem, which asserts that in the absence of critical values, the homology remains unchanged provided the dynamics on the partially ordered set satisfies a certain homological condition. This result, combined with the adjunction theorem, sufficed to prove a strengthening of the Morse inequalities. The second collapsing theorem guaranteed that, in the absence of critical values, the weak homotopy type remains unchanged, provided the dynamics on the partially ordered set satisfies an extra condition. Finally, we extended the Morse inequalities by taking torsion into account. Acknowledgments The authors would like to thank an anonymous referee for both fixing and improving Proposition 4.7 and its proof. The first and the fifth authors were partially supported by MICINN Spain Research Project PID2020-118753GB-I00 and Junta de Andaluc´ ıa Research Groups FQM–326 and FQM–189. The third author was partially 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 authors were partially suported by MICINN Spain Research Project PID2020-114474GB-I00. Conflict of interest We declare no conflict of interest. AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14944 References 1. P. Alexandroff, Diskrete R¨ aume, Rec. Math. Moscou, n. Ser.,2(1937), 501–519. 2. T. F. Banchoff, Critical points and curvature for embedded polyhedra, J. Differ. Geom.,1(1967), 245–256. https://doi.org/10.4310/jdg/1214428092 3. T. F. Banchoff, Critical points and curvature for embedded polyhedra. II. Differential geometry (College Park, Md., 1981/1982), 34–55, Progr. Math., 32, Birkh¨ auser Boston, Boston, MA, 1983. 4. J. A. Barmak, Algebraic topology of finite topological spaces and applications, Lecture Notes in Mathematics 2032, Springer, Heidelberg, 2011. 5. J. A Barmak, E. G. Minian, One-point reductions of finite spaces, h-regular CW-complexes and collapsibility, Algebr. Geom. Topol.,8(2008), 1763–1780. https://doi.org/10.2140/agt.2008.8.1763 6. J. A. Barmak, E. G. Minian, Automorphism groups of finite posets, Discrete Math.,309 (2009), 3424–3426. https://doi.org/10.1016/j.disc.2008.09.026 7. J. A. Barmak, E. G. Minian, Strong homotopy types, nerves and collapses, Discrete Comput. Geom.,47 (2012), 301–328. https://doi.org/10.1007/s00454-011-9357-5 8. M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math.,129 (1997), 445–470. https://doi.org/10.1007/s002220050168 9. E. D. Bloch, Polyhedral representation of discrete Morse functions, Discrete Math.,313 (2013), 1342–1348. https://doi.org/10.1016/j.disc.2013.02.020 10. K. S. Brown, Euler characteristics of groups: The p-fractional part, Invent. Math.,29 (1975), 1–5. https://doi.org/10.1007/BF01405170 11. M. K. Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math.,217 (2000), 101–113. https://doi.org/10.1016/S0012-365X(99)00258-7 12. P. J. Chocano, M. A. Mor´ on, F. Ruiz del Portal, Topological realizations of groups in Alexandroff spaces, Rev. R. Acad. Cienc. Exactas F´ıs. Nat., Ser. A Mat., RACSAM,115 (2021), Id/No 25. https://doi.org/10.1007/s13398-020-00964-7 13. N. Cianci, M. Ottina, Poset splitting and minimality of finite models, J. Comb. Theory, Series A, 157 (2018), 120–161. https://doi.org/10.1016/j.jcta.2018.02.010 14. F. D. Farmer, Cellular homology for posets, Math. Japon.,23 (1978/79), 607–613. 15. R. Forman, Morse theory for cell complexes, Adv. Math.,134 (1998), 90–145. https://doi.org/10.1006/aima.1997.1650 16. R. Forman, Witten-Morse theory for cell complexes, Topology,37 (1998), 945–979. https://doi.org/10.1016/S0040-9383(97)00071-2 17. R. Forman, A user’s guide to discrete Morse theory, S´em. Lothar. Combin,48 (2002), Art. B48c, 35 pp. 18. R. Ghrist, Elementary Applied Topology, Createspace, 2014. 19. K. A. Hardie, J. J. C. Vermeulen, P. J. Witbooi, A nontrivial pairing of finite t0 spaces, Topol. Appl., 125 (2002), 533–542. https://doi.org/10.1016/S0166-8641(01)00298-X AIMS Mathematics Volume 7, Issue 8, 14922–14945. 14945 20. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. 21. A. T. Lundell, S. Weingram, The topology of CW complexes, The University Series in Higher Mathematics, Van Nostrand Reinhold Co., New York, 1969. 22. M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J.,33 (1966), 465–474. https://doi.org/10.1215/S0012-7094-66-03352-7 23. J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963. 24. E. G. Minian, Some remarks on Morse theory for posets, homological Morse theory and finite manifolds, Topology Appl.,159 (2012), 2860–2869. https://doi.org/10.1016/j.topol.2012.05.027 25. M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ. 18 AMS, Providence, RI, 1996. 26. E. Pitcher, Inequalities of critical point theory, Bull. Am. Math. Soc.,64 (1958), 1–30. https://doi.org/10.1090/S0002-9904-1958-10137-8 27. D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math.,28 (1978), 101–128. https://doi.org/10.1016/0001-8708(78)90058-0 28. G. Raptis, Homotopy theory of posets, Homology Homotopy Appl.,12 (2010), 211–230. https://doi.org/10.4310/HHA.2010.v12.n2.a7 29. R. E. Stong, Finite topological spaces, Trans. Amer. Math. Soc.,123 (1966), 325–340. https://doi.org/10.1090/S0002-9947-1966-0195042-2 30. R. E. Stong, Group actions on finite spaces, Discrete Math.,49 (1984), 95–100. https://doi.org/10.1016/0012-365X(84)90156-0 31. M. L. Wachs, Poset topology: Tools and applications, In: Geometric combinatorics, 497–615. IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007. ©2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0) AIMS Mathematics Volume 7, Issue 8, 14922–14945.