Fundamental Theorems of Morse Theory on posets

We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or 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 the context of finite spaces.

The first and the fifth authors were partially supported by MINECO Spain Research Project MTM2015-65397-P and Junta de Andalucía Research Groups FQM-326 and FQM-189. The second author was partially supported by MINECO-FEDER research project MTM2016-78647-P. The third author was partially supported by Ministerio de Ciencia, Innovación y Universidades, grant FPU17/03443. The second and third authors were partially supported by Xunta de Galicia ED431C 2019/10 with FEDER funds.
The organization of the paper is as follows. In Section 2 we recall some definitions and standard results about posets or finite topological spaces. Section 3 is devoted to the study of discrete Morse Theory in the context of posets. In Section 4 we prove the Fundamental Theorems of Morse Theory in this setting. Finally, in Section 5 we show some of their consequences, such us extending original Forman's result regarding the homotopy type of a regular CW-complex with a Morse function defined on it or obtaining the Morse inequalities. Moreover, we study a method to reduce the criticality of a Morse function defined on a poset.

Finite Spaces, posets, and simplicial complexes
This section is devoted to introduce the objects we will work with. In particular we are interested in two kinds of posets, two-wide posets and down-wide posets (the first of which is due to Bloch [9]), for which we will establish the main results of this work. Most of the material presented is well established in the literature, for further details the reader is referred to [4,6,9,11,20,25]. All posets will be assumed to be finite and by finite space we will mean T 0 -space.

Preliminaries.
It is well known that finite posets and finite T 0spaces are in bijective correspondence. If (X, ≤) is a poset, a topology on X is defined by taking as a basis the open sets U x := {y ∈ X : y ≤ x} for each x ∈ X. On the other hand, if X is a finite T 0 -space, for any x ∈ X, define the minimal open set U x as the intersection of all open sets containing x. Then X may be given a poset structure by defining y ≤ x if and only if U y ⊆ U x . It is easy to see that these correspondences are mutual inverses of each other. Moreover a map between posets f : X → Y is 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 T 0 -spaces, we will use both notions interchangeably.
We need to introduce some basic notions and results.
Definition 2.1.1. 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 height of X is the maximum length of the chains in X, where a chain x 0 < x 1 < . . . < x n has length n. The height h(x) of an element x ∈ X is the height of U x with 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 U x is 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, that is, 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 U x := {w ∈ X : w < x} as well as F x := {y ∈ X : y ≥ x} and F x := {y ∈ X : y > x}.

2.2.
Beat points and γ−points. Due to the correspondence between posets and finite topological spaces we can study the homotopy type of a poset.
Notice that in both, U x and F x are contractible.
The next proposition states that removing beat points from a poset does not change its homotopy type. There is a weaker notion of beat point which we recall now: McCord Theory of weak equivalences. We now recall Mc-Cord functors between posets and simplicial complexes. Given a poset X, we define its order complex K(X) as the simplicial complex whose k-simplices are the non-empty chains of X of length k. Furthermore, given an order preserving map f : X → Y between posets, we define a simplicial map K(f ) : Conversely, if K is a simplicial complex, we define the face poset of K, denoted ∆(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) by ∆(φ)(σ) = φ(σ) for each simplex σ of K.
The face poset functor is defined analogously for CW-complexes. That is, given a CW-complex K, ∆(K) is the poset of cells of K ordered by inclusion. Given a cellular map φ : K → L we define the order preserving map ∆(φ) : ∆(K) → ∆(L) by ∆(φ)(σ) = φ(σ) for each cell σ of K. Note that for a simplicial complex K, K∆(K) is sd(K), the first barycentric subdivision of K. For details and a proof of the result below see [4]: The following statements hold: (1) Let X be a finite T 0 -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.
In the proof of Theorem 2.
is a weak homotopy equivalence. Then f : X → Y is a weak homotopy equivalence.
2.4. Two-wide posets. In this subsection we introduce a class of posets important for the later development of Morse Theory.
Definition 2.4.1 ([9]). A poset X is two-wide if for any x, z, y such that x ≺ z ≺ y, there is some z ′ ∈ X such that z ′ = z and x ≺ z ′ ≺ y.
Remark 1. A poset X is two-wide if and only if it satisfies the following condition: for any pair of elements x, y ∈ X such that x < y and x ⊀ y, #{z : 2.5. Down-wide posets. In this subsection we introduce the new concept of down-wide poset, which will play an important role in the development of Morse Theory in the context of finite posets.
Definition 2.5.1. Given a poset X and x ∈ X, we define the downincidence number of x as the cardinality of the set ∂(x) = {y ∈ X : y ≺ x}. The poset X is down-wide if #∂(x) ≥ 2 for every for every nonminimal x in X.
Obrserve that down-wide posets do not have down-beat points. It is easy to check the following result:    Observe that a subposet Y of a poset X is an open subposet (seeing both posets as finite spaces) if whenever x ∈ Y and y ≤ x, then y ∈ Y . Therefore, an open subposet of a down-wide poset is down-wide.
Observe that the properties of being two-wide and down-wide do not imply each other.
2.6. (Homologically) admissible posets. We recall the notion of (homologically) admissible posets introduced by Minian [20]. We denote by H(X) the Hasse diagram associated to the poset X. In order to check if an edge (w, x) ∈ H(X) is admissible, one needs to compute the higher homotopy groups of U x −{w}, which is a difficult problem. That is why we introduce the following weaker notion: We also recall the notion of homologically admissible poset, which is weaker than admissible poset. Recall that given a poset X, since it is also a finite topological space, its singular homology is defined.  We present below an important class of examples of homologically admissible posets. For a detailed exposition the reader is referred to [8,20,22]). Definition 2.6.4 ([20]). A simplicial complex K is a closed homology manifold of dimension n if the link of every simplex has the homology of S n−k−1 , where k is the dimension of the simplex. A poset X is a finite closed homology manifold if its order complex K(X) is a closed homology manifold. Lemma 2.6.5. [20, Theorem 4.6] If X is a finite closed homology manifold, then it is homologically admissible.
As a consequence of this result, the theory developed for homologically admissible manifolds can be applied to study the topology of triangulable homology manifolds by means of their order triangulations.
We end the subsection by relating the properties of being (homologically)-admissible with those of being two-wide and down-wide. First, it is easy to check the flowing lemma: Lemma 2.6.6. Let X be a poset. If X is homologically admissible, then it is down-wide.
Remark 4. In Lemma 2.6.6 it is assumed that the empty set is not acyclic.
Second, the cellular homology developed by Minian in [20] allows to adapt the ideas of [12, Theorem 1.2 (iii)] so the proposition below follows easily: Proposition 2.6.7. Let X be a poset. If X is homologically admissible, then it is two-wide.

Morse Theory on posets
3.1. Definition of Morse functions. We recall the definition of Morse function for posets introduced by Minian [20]. It is an adaptation of Forman's theory [14,12] to the context of posets.
The set of critical points is denoted by critf . The images of the critical points are called critical values. The points (values) which are not critical are said to be regular points (regular values).

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 [12, 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 so it is left for the reader.

Lemma 3.2.1 (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: Remark 5. Observe that Lemma 3.2.1 does not hold in general for finite spaces. As an example, consider the poset of Figure 3.1 taking the points labeled as z and y.
Definition 3.2.2. Given a poset X, a Morse function f : X → R is 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), The following result plays the role, in the context of finite spaces, of the Exclusion Lemma [12, Lemma 2.5]. Proof. First of all, since x is not critical, then at least one of the conditions holds. We will see that the conditions are mutually exclusive. So, assume both conditions hold and we will arrive to a contradiction. By Condition 2, x is not a minimal element. By Lemma 3.2.1 there exists x ′ such that: As a consequence we obtain the following chain of inequalities: 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.

Matchings.
We recall the definition of matching introduced to the context of discrete Morse theory by Chari [10] and further developed by Minian [20].
• each x ∈ X belongs to at most one element in M.     Figure  3.1 is homotopy-regular, and therefore homology-regular, since for every x (p) ∈ crit(M), the subspace U x has the homotopy type of a finite model of S p−1 .
Minian proved an integration result for matchings which can be stated as follows.
As a consequence of our Exclusion Lemma for Morse functions on two-wide posets (Lemma 3.2.3), we obtain a converse result.  Therefore, we can establish a correspondence between Morse matchings and order preserving Morse functions satisfying the Exclusion condition on graded posets. However, the correspondence is not bijective since given a Morse function f : X → R, the function f ′ : X → R given by f ′ (x) = 2f (x) is again Morse and both functions share the same associated matching.

Fundamental Theorems
4.1. First observations. We introduce the following notation: given a finite poset X and a Morse function f : Observe that, for each a ∈ R, the subposet X a is an open subset of X. We denote by b 0 (X) the number of connected components of X, which coincides with the number of path-connected components. Given a discrete Morse function on a simplicial complex f : K → R, it holds that, as a ∈ R increases, new connected components of K a arise as critical vertices. The following result asserts this phenomenon for downwide posets.
, then there exists a critical value c ∈ (a, b] such that b 0 (X c ) = b 0 (X a ) + 1. Moreover, the critical value c corresponds to a minimal element of X.
Proof. Since f is injective, then the number of path-components of X t can only increase by one when t reaches a new regular or critical value. Denote by c ′ the minimum value among the critical or regular values which are strictly greater than a and such that b 0 (X c ′ ) = b 0 (X a )+1. Let us denote the element corresponding to the value c ′ by x, i.e. f (x) = c ′ .
We have to prove that x is critical and that x is minimal so c = c ′ is the claimed value.
Let us begin by proving that x is minimal. Assume that x is not minimal and we will arrive to a contradiction. Since we are adding a new path-component at c ′ , all the elements of ∂x must satisfy that their values by f are strictly greater than f (x) = c ′ . Moreover, by hypothesis, #∂x ≥ 2, which is a contradiction with the fact that f is a Morse function. Now, we prove that c ′ must be a critical value. We argue by contradiction again. If x is not a critical element, since x is minimal, then there exists y ≻ x such that f (y) < f (x). Therefore, x ∈ X f (y) . Furthermore, applying that #∂y ≥ 2 and the definition of Morse function to y, there exists x ′ ≺ y such that f (y) > f (x ′ ). This is a contradiction with the fact that we are adding a new path-component when we reach c ′ .   With the following result we prove that the addition of regular elements does not create new connected components.     This subsection is devoted to proving the substitutes of the fundamental theorems of Morse Theory in this context, that is, two collapsing theorems and an adjunction theorem. The first collapsing theorem is a homological collapsing theorem, which asserts that, in the absence of critical values, the homology remains unchanged provided the matching is homologically admissible. This result, combined with the adjunction theorem, is enough to prove the Morse inequalities. The second collapsing theorem guarantees that, in the absence of critical values, the weak homotopy type remains unchanged, provided that the matching is 1-admissible and homologically admissible. This result is analogous to [12,   In order to prove Theorem 4.2.2 we begin with the following easy lemma which allow us to perturb the function locally so it can be taken to be injective. (1) X g t = X f t for every t < c. (2) The restriction of g to (a, c] is injective. As a consequence of Lemma 4.2.3, in the next proofs we will assume the injectivity of certain functions. Proof. Assume that f is injective and that furthermore, (a, b] contains only one regular value c = f (v), then a < c < b. Since v is a regular point and f satisfies the Exclusion condition, Proposition 4.1.3 implies that we only need to consider the following two mutually exclusive cases: where w ≺ v and w is an up beat point. In order to prove the claim, suppose there exists u = v, such that w ≺ u.
Claim. Under the above conditions, If u ∈ X b , then u ∈ X a .
Proof of the Claim. (a) First, since f (v) < f (w) and v is the unique regular element in (a, b], we must have b < f (w) < f (u) by the definition of a Morse function. So, suppose there exists u = v, such that w ≺ u. By the Claim there are two cases to consider: 1 u ∈ X b . Then u ∈ X a and w is a (up) beat point. 2 u / ∈ X b and, again, w is a (up) beat point.
Since w ≺ v is an element in the matching and the matching is homologically admissible, then U v −{w} is acyclic. By applying the Long Exact Sequence of homology to the pair (U v , U v −{w}) and using the fact that   Proof. We will apply McCord's Theorem (Theorem 2.3.2) to the base There are two cases to consider: has a maximum and therefore is con-   ((a, b]). Then Proof. We may assume that f is injective, that f (x) > a and that the only point in f −1 ((a, b]) is x. Since x is critical, then, given y (p+1) ≻ x, f (y) > f (x). Hence, f (y) > b and since f is order preserving and satisfies the Exclusion condition, then f (z) > b for every z > x. Therefore, x ∩ X a = ∅. Given any w (p−1) ≺ 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 (p) x (p) .

5.1.
Extension of Forman's Decomposition Theorem. As a first consequence, we extend Forman's Discrete Morse theory on regular CW-complexes to more general cell complexes. We recover Forman's result [12, Corollary 3.5] as a particular case. Moreover, we do not need to make use of simple homotopy types. We will work with less rigid cell structures than regular CW-complexes while maintaining some combinatorial structure. Given an h-regular CW-complex K, the cells whose attaching maps are not homeomorphisms are called irregular cells. For a detailed exposition of h-regular CW-complexes and some examples the reader is referred to [3,4].
We recall Forman's definition of Morse function on an h-regular CWcomplex K [12].
Definition 5.1.2. Let K be an h-regular CW-complex, a discrete Morse function on K is a map f : K → R such that, for every p-cell σ (p) ∈ K, we have: (1) If σ is an irregular face of τ (p+1) , then f (τ ) > f (σ). Moreover, (2) If β (p−1) is an irregular face of σ, then f (β) < f (σ). Moreover, We present a generalized notion of a discrete Morse function on a h-regular CW-complex K: Our definition generalizes Forman's since we do not force non-regular cells to be critical.
We recall from the proof of [4, Theorem 7.1.7] the following result, which is a generalization of Theorem 2.3.1 (2): Proposition 5.1.4. Let K be a finite h-regular CW-complex and let L ⊂ K be a subcomplex.
(1) There are maps f K : K → ∆(K) and f L : L → ∆(L) defined in [4,Theorem 7.1.7] which are weak homotopy equivalences. (2) The following diagram is commutative:  ((a, b]). Then the same combinatorial proof of [12,Theorem 3.4] (observe that here we are using that ∆(f ) : ∆(K) = X → R is order preserving and satisfies the Exclusion condition) proves that K b has the same homotopy type as K a with a p-cell attached. The result follows.
Remark 8. Corollary 5.1.5 provides an alternative approach to [20,Corollary 4.2] by means of our Fundamental Theorems. Moreover, it does not involve the use of simple homotopy type.

5.2.
Morse-Pitcher Inequalities. Another consequence of our structural theorems of Morse Theory for finite spaces is that we can reproduce the classical proof (see [23,19] for the standard argument) of Morse-Pitcher inequalities in this context.
We consider coefficients in a principal ideal domain. Let f : X → R be a Morse function, we denote by m i the number of critical points of height i and by b i the Betti number of dimension i without expliciting the space nor the domain of coefficients.
Corollary 5.2.1 (Strong Morse inequalities). Let X be a down-wide poset and let f : X → R be an order preserving Morse function satisfying the Exclusion condition. Suppose that the Morse matching associated to f is homologically admissible and homology-regular. Then, for every n ≥ 0 and domain of coefficients: m n − m n−1 + · · · + (−1) n m 0 ≥ b n − b n−1 + · · · + (−1) n b 0 .
Corollary 5.2.2 (Weak Morse inequalities). Let X be a down-wide poset and let f : X → R be an order preserving Morse function satisfying the Exclusion condition. Suppose that the Morse matching associated to f is homologically admissible and homology-regular. Then: (1) m i ≥ b i for every i.
(2) The Euler-Poincaré Characteristic satisfies Remark 9. The Morse inequalities for homologically admissible posets can also be derived by following a combinatorial Hodge-theoretic argument mimicking [13] since the arguments provided by Forman can be reproduced without changes in this context. Moreover, we also recover Pitcher strengthening of Morse inequalities. We denote by µ i the minimum number of generators of the torsion subgroup T i of H i (X).
Observe that if deg(X) = n, then µ n = 0 since H n (X) is a subgroup of the free abelian group C n (X). Moreover, µ 0 = 0 and µ −1 is defined as 0.

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 in canceling pairs of critical elements, extending to the context of posets known results on smooth manifolds and simplicial complexes. First, we introduce some terminology. Given a matching M on the poset X, we will decompose X as the disjoint union of three subsets X = crit(M) ⊔ s(M) ⊔ t(M). For each edge (x, y) ∈ M, we say that x is the source of the edge and y is the target. We define the source of the matching s(M) as set whose elements are the sources of the edges in the matching. Analogously, we define the target of the matching t(M) as 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) = x and t(x) = y.
r =x such that for each i ∈ {0, . . . , r − 1}: (1) (x i , y i ) ∈ M, We present a result, which can be seen as the adaptation of [12, Theorem 11.1] to our context. Theorem 5.3.2 (Canceling critical points). Given a matching M on a finite graded poset X, assume that z (p+1) and x (p) are critical points such that there is an element y (p) ≺ z (p+1) and an unique M-path γ : z ≻ y = x 0 ≺ z 0 ≻ x 1 ≺ z 1 ≻ · · · ≺ z r ≻ x r = x (there is no other M-path from any p-face of z (p+1) to x (p) ). Then there is a matching M ′ such that: • The set of critical points of M ′ is crit(M ′ ) = crit(M) − {x, z}.
• Moreover, M ′ = M except along the unique gradient path from ∂z to x.
Proof. We define M ′ as follows: (1) t M ′ (w) = t M (w) if w / ∈ {y, z 0 , x 1 , z 1 , . . . z r , x} (M ′ = M except along the unique gradient path from ∂z to x) (2) t M ′ (x i ) = z i−1 , i = 1, . . . , r (we reverse the gradient path from x to z 0 so x is no longer critical) (3) t M ′ (y) = z (we reverse the arrow from y to z so z is no longer critical). It remains to check that there are no closed M ′ -paths. We argue by contradiction. Suppose there was a closed M ′ -path δ.
Claim. Under the above hypothesis, δ would contain at least one pelement from γ and one p-element not in γ.
Proof of the Claim. The elements coming from γ can not give a closed M ′ -path on their own since we have just reverted their arrows. The elements of X which are not in γ can not give a closed M ′ -path since in that case we would also have a closed M-path and M is a gradient vector field. Therefore in δ we must have at least one element in each of their sets. Moreover, if we have a (p + 1) element, then we have a p-element, so we have at least one p-element of each of one of these sets.
Hence, δ would contain a sequence of the form: x i ≺ w 0 ≻ s 1 ≺ w 1 ≻ · · · ≺ w s ≻ x j where s ≥ 0, w i = x k , w i = z k , s i = x k , s i = z k , for all i and k. Since t M ′ (w i ) = t M (w i ) and t M ′ (s i ) = t M (s i ) for all i, we have a M-path: Let us consider two cases: (1) If i = 0, then s 1 = x i−1 , x i and s 1 ≺ t M ′ (x i ) = z i−1 . Therefore, we can define a second gradient M-path γ ′ = γ from ∂z to x: Which is a contradiction. (2) If i = 0, then y = x 0 = s 1 ≺ t M ′ (y) = z. Therefore, we can define the following M-path: γ ′ : z ≻ s 1 ≺ w 1 ≻ · · · ≺ w s ≻ x j ≺ z j ≻ · · · ≻ x r = x which is different from γ and also goes from ∂z to x. Then we have a contradiction.
Finally, there is a kind of dual result to Theorem 5.3.2 which allows us to create critical points. Both the statement and the proof are a straightforward translation of [12,Theorem 11.3].