Topological combinatorics: discrete Morse theory Emanuele Ventura

Introduction Topological combinatorics is a beautiful subject unifying two of the most ubiquitous subjects in mathematics: topology and combinatorics. Hence it is not surprising that this field is having and will have a huge impact in pure mathematics and applications. Robin Forman introduced discrete Morse theory, extending Morse theory on smooth manifolds to the setting of CW-complexes. Discrete Morse theory is one of the most powerful tools in topological combinatorics and it has been adopted successfully in the study of many simplicial complexes. The aim of this notes is to survey discrete Morse theory. We refer to the wonderful monograph by Jonsson [2] for complete proofs and other interesting applications of discrete Morse theory.

1

Background from algebraic topology

One of the first theorems connecting topology and combinatorics was proved by Quillen. Proposition 2 (Quillen Fiber Theorem). Let P and Q be posets. Let f : P → Q be a poset map. A poset map is a map respecting the order relations. If f is such that the order complex of f −1 (Q ≤q ) is contractible for all q ∈ Q, then the simplicial map ∆( f ) : ∆(P) → ∆(Q) is a homotopy equivalence.

3

Discrete Morse theory

We start this section reviewing the notion of elementary simplicial collapse. Definition 1 (Elementary simplicial collapse). Let ∆ be a simplicial complex. An elementary simplicial collapse is a removal of interiors of two simplices σ and τ such that 1

(i) dim σ = dim τ + 1; (ii) the only simplex containing σ is σ itself; (iii) the only simplices containing τ are τ and σ. These collapses are also called elementary collapses. A simplicial collapse is possible if and only if there exists a simplex τ whose link in ∆ consists of single vertex. In this case the simplex σ is given by the span of τ and σ. Indeed, if there exists such a simplex then we are done. Suppose that a simplicial collapse is possible. Then there exist two simplices σ and τ such that the dimension of τ is one less, the only simplex containing σ is σ and the only simplices containing τ are σ and τ. Consider the link τ. The link contains at least one vertex, since σ is in the generalized simplicial complex. Suppose that there is another vertex in the link, then this is a contradiction. The statement follows. Let ∆ be a simplicial complex. Let {σ, τ} be a pair of faces of ∆ such that σ ⊂ τ and dim σ = dim τ − 1. For an ordinary collapse we require τ to be a maximal face and the only face containing σ. In this situation we say that {σ, τ} is free in ∆. If we have not this condition we have a generalized elementary collapse. In the following, we describe geometrically a generalized collapse with respect to σ and τ. First we remove the open set kτk from k∆k and we identify k2σ k with k∂ 2τ \ σk. A generalized elementary collapse does not affect the homotopy type of ∆. Note that the resulting complex is not necessarily a simplicial complex, but it is a general CW-complex. We can have many generalized elementary collapses and combine them to obtain a generalized collapse. This operation will keep intact the homotopy type. This is the main principle of discrete Morse theory for the general case when ∆ is a CW-complex. We have a set of pairs {σ1 , τ1 }, . . ., {σ r , τ r } that we collapse. We can see these pairs as a matching on ∆. The faces contained in some matching will be called matched faces, otherwise we call them unmatched. Let ∆i−1 be the resulting cell complex after the first i − 1 generalized elementary collapses; set ∆0 = ∆. The rule for the pairs to form a sequence of ordinary elementary collapses is that each pair {σi , τi } is free in ∆i−1 . For generalized elementary collapses, we have the same rule but restricted to matched faces. More specifically, we do not require that {σi , τi } to be free in ∆i−1 , but τi must be the only matched face of ∆i−1 containing σi . This means that for each i, we require that σi is not contained in τi+1 , . . . , τ r . A matching on ∆ admitting an order with this property is called acyclic. Example 4. Consider the triangle on vertices a,b and c. Consider the following matching {{a, a b}, {b, bc}, {c, ac}}. Any order of these three pairs does not give in any case a sequence of generalized elementary collapses. Remark 1. The main result of discrete Morse theory is that an acyclic matching induces a homotopy equivalence between ∆ and ∆ r , the complex resulting from the sequence of generalized elementary collapses. Then ∆ is homotopy equivalent to a cell complex with as many 2

cells of dimension d as there are unmatched faces of ∆ of the same dimension. Since the number of cells in each dimension is an upper bound for the rank of the homology group in that dimension, then this procedure gives upper bounds on the ranks of homology groups of ∆. If all the unmatched cells are of the same dimension then ∆ r is a wedge of d-dimensional spheres.

5

Acyclic matchings

We consider acyclic matchings on families of sets. Definition 2. Let X be a set and let ∆ be a finite family of finite subsets of X . A matching on ∆ is a family M of pairs {σ, τ} with σ, τ ∈ ∆ such that no set is contained in more than one pair in M . A set σ in ∆ is critical or unmatched with respect to M if σ is not contained in any pair in M . We say that a matching on M on ∆ is an element matching if every pair in M is of the form {σ − x, σ + x} for some x ∈ X and σ ⊆ X . Consider an element matching M on a family ∆. Let D = D(∆, M ) be the direct graph with vertex set ∆ and with a directed edge from σ to τ if and if one of the following holds: (i) {σ, τ} ∈ M and τ = σ + x for some x ∈ / σ; (ii) {σ, τ} ∈ / M and σ = τ + x for some x ∈ / τ. Thus every edge in D is an edge in the Hasse diagram of ∆ ordered by inclusion. Edges corresponding to pairs of matched sets are directed from the smaller to the bigger one. All the other edges are directed the other way around. We write σ → τ if there is a directed path from σ to τ in D. Definition 3 (Acyclic matchings). An element matching M is an acyclic matching if D is acyclic; equivalently, σ → τ and τ → σ implies that σ = τ. We prove that any cycle in a directed graph D corresponding to an element matching is of the form {{σ0 , τ0 }, {σ1 , τ1 }, . . . , {σ r−1 , τ r−1 }} with r > 1 such that σi , σ(i+1) mod r ⊂ τi and {σi , τi } ∈ M . Proof. Suppose we have a cycle in D(∆, M ). The cycle is given by elements of the matching {{σ0 , τ0 }, {σ1 , τ1 }, . . . , {σ r−1 , τ r−1 }}. The cycle starts and ends in σ0 . From σ0 going along the Hasse diagram we arrive at τ0 . By definition of matching τ0 is not allowed to be in the pairs of the matching more than one time. Then we can go below from the larger to the smaller σ1 , since it is matched only with τ1 . This holds in every step until we arrive at τ r−1 for some r and then the last edge of the cycle is from τ r−1 to σ0 . This is precisely the claimed type of cycle.

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Lemma 6. Let ∆ ⊆ 2X and x ∈ X . Let M x = {{σ − x, σ + x}|σ − x, σ + x ∈ ∆} and ∆ x = {σ|σ − x, σ + x ∈ ∆}. Let M 0 be an acyclic matching on ∆0 := ∆ \ ∆ x . Then M := M x ∪ M 0 is an acyclic matching on ∆. Proof. Assume that (σ0 , τ0 , . . . , σ r−1 , τ r−1 ) is a cycle in D(∆, M ) satisfying the conditions proved above in Definition 3. Since M 0 is an acyclic matching on ∆0 , there must be a pair (σi , τi ) included in M x ; by construction τi = σi + x. Assume i = 0 for simplicity. Since τ r−1 is not matched with σ0 then x ∈ / τ r−1 . This means that there is some j ∈ [1, r − 1] such that x ∈ τ j−1 and x ∈ / τ j . On the other hand, this implies that τ j−1 = σ j + x, which is a contradiction, because otherwise we would have (σ j , τ j−1 ) ∈ M x by construction. Lemma 7 (Cluster Lemma or Patchwork Lemma). Let ∆ ⊆ 2X and let f : ∆ → Q be a poset map, where Q is any poset. For q ∈ Q, let Mq be an acyclic matching on f −1 (q). Let [ M= Mq . q∈Q

Then M is an acyclic matching on ∆. Proof. Suppose (σ0 , τ0 , . . . , σ r−1 , τ r−1 ) is a cycle in D(∆, M ). Let q0 , . . . , q r−1 be distinct elements in the poset Q such that σk , τk ∈ f −1 (qk ) for 0 ≤ k ≤ r − 1. Since σ(k+1) mod r ⊂ τk then q(k+1) mod r = f (σ(k+1) mod r ) ≤ f (τk ) = qk . This implies by induction that qk0 ≤ qk for any pair k, k0 . Swapping k and k0 we obtain qk = qk0 , since Q is a poset. All the sets in the cycle are contained in one single family f −1 (q), a contradiction with the choice of qk ’s. Remark 2. If no cell in the acyclic matching is covered by a critical face, then the Morse complex is a simplicial complex and the homotopy equivalence is a deformation retraction. Lemma 8. Let ∆0 and ∆1 be disjoint families of subsets of a finite set such that τ * σ if σ ∈ ∆0 and τ ∈ ∆1 . (This latter condition means that the sets in the disjoint families are incomparable in the Hasse diagram.) If Mi is an acyclic matching on ∆− i for i = 0, 1, then M0 ∪M1 is acyclic matching on ∆0 ∪ ∆1 . Proof. Using the Lemma 7 and choosing Q to be the poset given by two elements q1 and q2 such that ∆1 ⊂ f −1 (q1 ) and ∆2 ⊂ f −1 (q2 ), we have the statement.

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9

Simplicial Morse theory

Let ∆ be a simplicial complex with at least one vertex and let M be an acyclic matching on ∆. We can assume that the empty set is in the matching. Indeed, suppose that all the 0-cells – the vertices – are matched with larger faces, then there is a cycle in the directed graph D(∆, M ); this comes from the consideration on the directions of edges between vertices and larger faces. Note that many of the results of Forman’s discrete Morse theory apply to CW-complex, but here we stick with simplicial complexes for the sake of simplicity. Theorem 10 (Forman). Suppose that ∆0 is a subcomplex of ∆ such that ∆0 9 ∆ \ ∆0 and such that all critical faces belong to ∆0 . Then it is possible to collapse ∆ to ∆0 , In particular, ∆ and ∆0 are homotopy equivalent. Hence ∆ has no homology in dimensions strictly greater than dim ∆0 . Proof. The restriction of the acyclic matching to ∆ \ ∆0 is a perfect matching. Namely, if τ ∈ ∆ \ ∆0 is matched with σ ∈ ∆0 , then σ ⊂ τ, which implies that σ → τ, which is a contradiction. We use induction on |∆ \ ∆0 | to prove the claim. If ∆ = ∆0 , then we are done. Let σ be a face of ∆ \ ∆0 such that no edge in the directed graph D corresponding to the matching ends in σ; such a face exists by the acyclicity of D and by assumption that ∆0 9 ∆ \ ∆0 . The face σ is matched with a larger face τ and that σ is not contained in any other face. In particular, we can collapse ∆ to the subcomplex ∆ \ {σ, τ}. By induction, we can collapse ∆ \ {σ, τ} to ∆0 . This proves the statement. Proposition 11. Let ∆ be a simplicial complex and let x be a 0-cell in ∆. Let y be a new 0-cell and define ∆0 to be the complex obtained from ∆ by adding σ + y and σ + x + y whenever σ + x ∈ ∆. Then ∆ and ∆0 are homotopy equivalent. Proof. Note that ∆0 is the disjoint union of ∆ and Del t a00 = {{ y}} ∗ ∆(x), where ∆(x) is the subcomplex of ∆ given by all the faces in ∆ containing x. By Lemma 8, acyclic matchings on these two families of sets give an acyclic matching on ∆0 . We define a matching on ∆00 by pairing σ − x with σ + x. This is a perfect acyclic matching, which gives an acyclic matching on ∆0 such that a face is critical if and only if the face is in ∆. By Theorem 10, we have that ∆0 collapses to ∆. Corollary 12. If ∆ does not contain any critical face, then ∆ is collapsible and hence contractible to a point. Remark 3. Note that collapsible implies contractible. The converse is not true.There are contractible spaces which are not collapsible. For example the dunce hat is contractible but not collapsible. 5

Theorem 13 (Forman). If all critical faces of ∆ have dimensio at least d, then ∆ is (d − 1)connected. Proof. Let ∆0 be the subcomplex of ∆ consisting of all faces of dimension less than d plus all faces of dimension d that are matched with smaller faces. Since the acyclic matching restricts to a perfect matching on ∆0 , the latter is contractible and hence (d − 1)-connected. Since ∆0 contains the (d − 1)-skeleton of ∆, this proves the claim. Theorem 14 (Forman). If all critical faces of ∆ are of the same dimension d, then ∆ is homotopy equivalent to a wedge of spheres of dimension d. Proposition 15. Let X be a non-empty finite set. Then sd(∂ 2X ) admits an acyclic matching with one unmatched face of dimension |X | − 2. As consequence, sd(∂ 2X ) is homotopy equivalent to a sphere of dimension |X | − 2. For an acyclic matching M on a simplicial complex ∆, let U (∆, M ) be the family of critical faces of ∆ with respect to M . For a family V ⊆ U (∆, M ), let ∆V = {σ ∈ ∆|V → σ} ∪ {;, {x}}, where {x} is the 0-face matched with the empty set in M . If V is nonvoid, then ∆V = {σ ∈ ∆|V → σ}. Lemma 16. The family of sets ∆V is a simplicial complex. That is, if {σ, τ} ∈ M with σ ⊂ τ and τ ∈ ∆V , then σ ∈ ∆V . In particular, U (∆V , MV ) = ∆V ∩ U (∆, M ), where MV is the restriction of M to ∆V . Theorem 17. Suppose that V ⊆ U = U (∆, M ) has the property that U \ V 9 V and V 9 U \ V . Then ∆ is homotopy equivalent to ∆V ∨ ∆U \V . In particular, ∆ is homotopy equivalent to ∆U . Corollary 18. If U is the disjoint union of families V1 , . . . , V r with the property that Vi 9 V j if Wr i 6= j, then ∆ is homotopy equivalent to i=1 ∆Vi . Corollary 19. Let V ⊆ U = U (∆, M ) be such that U \ {V } 9 V and V 9 U \ {V } for every V ∈ V . Then ∆ is homotopy equivalent to _ ( S |V |−1 ) ∨ ∆U \V . V ∈V

Theorem 20 (Forman; Funtamental Theorem of Simplicial Morse Theory). Let ∆ be a simplicial complex and let M be an acyclic matching on ∆ such that the empty set is not critical. Then ∆ is homotopy equivalent to a cell complex with one cell of dimension p ≥ 0 for each critical face of ∆ of dimension p plus one additional 0-cell. 6

The resulting complex provided by the sequence of generalized elementary collapses is the discrete Morse complex of ∆ with respect to M . We present the weak Morse inequality. They consequence of the Fundamental Theorem 20 and the isomorphism between simplicial and cellular homology. Theorem 21 (Forman). Let K be a field, let ∆ be a simplcial complex and let M be an acyclic ˜ d (∆; K) for matching on ∆. Then the number of critical faces of dimension d is at least dim H each d ≥ −1.

22 22.1

Families of complexes Contractible complexes

Let F be a field or the ring of integers Z. A simplicial complex ∆ is acyclic over F or F-acyclic if ∆ has no reduced homology over F. By the Universal Coefficient Theorem of homology, a complex ∆ is Z-acyclic if and only if is F-acyclic for any field F. Recall that for any field F there exists an F-acyclic complex that is not Z-acyclic. Example 23. Any triangulation of the real projective plane RP2 is F-acyclic whenever F is a field of characteristic different from 2, but it is not Z2 -acyclic nor Z-acyclic. ˜ d (∆; F) vanishes for d ≤ k. If A complex ∆ is called k-acyclic over F if the homology group H a complex ∆ is k-acyclic over Z then it is k-acyclic over any field F, but the converse it is not true for k ≥ 1. Proposition 24. Let d1 , d2 ≥ 0. If ∆ is (d1 − 1)-acyclic over F and Γ is (d2 − 1)-acyclic over F, then ∆ ∗ Γ is (d1 + d2 )-acyclic over F. Definition 4 (Contractible complexes). A simplicial complex ∆ is contractible if ∆ is homotopy equivalent to a point. Remark 4. A contractible complex ∆ is acyclic over Z. The converse is not true unless ∆ is simply connected; an example when this is not true is the Poincaré homology 3-sphere. (An homology k-sphere is a topological manifold with the same homology of the k-sphere.) Definition 5. Let k be a non-negative integer. A topological space X is k-connected if for every 0 ≤ d ≤ k, every continuous map f : Sd → X can be extended to a continuous map g : B d+1 → X . By convention X is −1-connected if it is non-empty. A topological space X is 0-connected if and only if it is path-connected. A space X is 1-connected if and only if it is simply connected. The connectivity degree of X is the largest integer k such that X is k-connected. We set +∞ if X is k-connected for every k ≥ 0.

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Theorem 25. For k ≥ 1, a simplicial complex ∆ is k-connected if and only if ∆ is k-acyclic over Z and simply connected. A simplicial complex ∆ is contractible if and only if ∆ is acyclic over Z and simply connected. For k ∈ {0, 1}, a simplicial complex ∆ is k-connected if and only if ∆ is k-acyclic. Corollary 26. For k ≥ 0, if ∆1 and ∆2 are k-connected and ∆1 ∩ ∆2 is (k − 1)-connected, then ∆1 ∪ ∆2 is k-connected. Corollary 27. If ∆ is a k-connected subcomplex of Γ and the dimension of each face of Γ \ ∆ is at least k + 1, then Γ is k-connected. Theorem 28. If ∆ is connected and dim Γ ≥ 0 (i.e., Γ is (−1)-connected), then ∆ ∗ Γ is simply connected (i.e., 1-connected). Corollary 29. Let d1 , d2 ≥ 0. If ∆ is (d1 − 1)-acyclic over Z and Γ is (d2 − 1)-acyclic over Z, then ∆ ∗ Γ is (d1 + d2 )-connected. Theorem 30. Let d ≥ 0. If ∆ is (d −1)-connected and dim ∆ ≤ d, then ∆ is homotopy equivalent to a wedge of spheres of dimension d.

30.1

Collapsible complexes

A complex is collapsible if the complex is void or can be collapsed to a point {;, {v}}. Collapsible complexes are contractible, but not all contractible complexes are collapsible. Definition 6. We define the class of collapsible simplicial complexes recursively as follows: (i) The void complex ; and any 0-simplex {;, {v}} are collapsible; (ii) If ∆ contains a non-empty face σ such that the face-deletion fdel ∆ (σ) and the link link ∆ (σ) are collapsible, then ∆ is collapsible.

30.2

Nonevasive complexes

Definition 7. We define the class of nonevasive simplicial complexes recursively as follows: (i) The void complex ; and any 0-simplex {;, {v}} are nonevasive; (ii) If ∆ contains a 0-cell x such that del ∆ (x) and link ∆ (x) are nonevasive, then ∆ is nonevasive. Example 31. Cones are nonevasive. Nonevasive complexes are contractible, but the converse it is not true.

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32

Independence complexes and discrete Morse theory

This section is based on a paper by Engström [1]. For a simplicial complex Σ choose a subset D of its vertex set. Consider the map f : F (∆) → F (Σ[D]) defined by σ 7→ σ ∩ D and use certain acyclic matchings on f −1 (τ) = {σ ∈ Σ|σ ∩ D = τ} ⊆ F (σ) to obtain an acyclic matching on all of F (∆).

Definition 8. A subset I of the vertex set of a graph G is independent if there are no two vertices of I which are adjacent in G. The independence complex of a graph G is a simplicial complex with the same vertex set and whose faces are given by the independent sets of G. The independence complex of G is denoted by I nd(G). Often it is used the fact that if v is an isolated vertexof G, then one obtains a complete acyclic matching on F (I nd(G)) by matching each σ which does not contain v with σ ∪ {v}. Lemma 33. If G is a graph with no distince vertices u and v which satisfy N (u) ⊆ N (v), then every acyclic matching on F (I nd(G \ {v})) can be extended to an acyclic matching on F (I nd(G)) with no new critical cells. Proof. Consider the map f : F (I nd(G)) → Q = {;, v} given by f (σ) = σ ∩ {v}. Note that f −1 (;) is F (I nd(G) \ {v}), where we have already an acyclic matching. We want to construct an acyclic matching on f −1 ({v}) which is the set of all independent sets containing v. Note that N (v) is not in these independent sets since, if w ∈ N (v) was contained in some independent set in f −1 (;), this would be a contradiction. Since N (u) ⊆ N (v) the same is true for the elements of N (u). If σ ∈ f −1 (;) then σ \ {v} ∈ f −1 (;). Then we have a complete matching in f −1 (;) given by {{σ, σ ∪ {u}| σ ∈ f −1 (;)}. Lemma 34. If G is the disjoint union of n > 0 edges then there is an acyclic matching on F (I nd(G)) with one critical cell. Proof. The proof is by induction on n. If n = 1 and V (G) = {u, v} then the acyclic matching {{;, {u}}} has one critical cell. If n > 1 and uv is an edge of G, then consider the poset map f : F (I nd(G)) → 2{v} defined by f (σ) = σ∩{v}. The subposet f −1 (;) is F (I nd(G\{v})) which has the isolated vertex u, and thus gives a complete acyclic matching. From the subposet f −1 ({v}) there is a poset bijection to F (I nd(G \ {u, v})) by removing v, and by induction we have an acyclic matching on F (I nd(G \ {u, v})) with one critical cell. Patching f −1 (;) and f −1 ({v}) together gives one critical cell. Proposition 35. If G is a forest, then there is an acyclic matching on F (I nd(G)) with either zero or one critical cell.

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Proof. The proof is by induction on the number of edges of G. If the number of edges is zero then G has an isolated vertex; in this case we have an acyclic matching with no critical cells. If G is a collection of disjoint edges, then by Lemma 34, there is an acyclic matching with one critical cell. Otherwise, there is a vertex u ∈ V (G) of degree one sitting in a connected component with more than two vertices. In this case there is a vertex v ∈ V (G) at distance two from u, which satisfies N (u) ⊆ N (v). By Lemma 33, we can extend any acyclic matching on F (I nd(G \ {v})) on F (I nd(G)) without introducing new critical cells. By induction, there is an acyclic matching on F (I nd(G \ {v})) with either one or zero critical cells. This proves the claim.

Theorem 36. X

˜ i (I nd(G); Q) ≤ dim H

i

min |I nd(G[D])|. is a forest

;6= D⊆V (G),G\D

Proof. Let D be a subset of V (G) of size φ(G), such that G \ D is a forest. If we remove even S more vertices from G it will still be a forest. In particular for any L ⊆ D, G \ (D ∪ v∈L N (v)) is a forest. We prove that there is an acyclic matching on I nd(G) with at most |I nd(G[D])| critical cells. Consider the poset map f : F (I nd(G)) → F (I nd(G[D]) defined by f (σ) = σ∩ D. We have divided the poset into |I nd(G[D])| subposets, and the next step is to show that each of them has at most one critical cell under some acyclic matching. For any L ⊆ D we have a poset bijection [ λ : F (I nd(G \ (D ∪ N (v)))) → f −1 (L) v∈L

given by λ(σ) = σ∪ L. By Proposition 35, there is an acyclic matching on f −1 (L) with at most S one critical cell, since G \ (D ∪ v∈L N (v)) is a forest. By the Cluster Lemma 7, we can patch these acyclic matchings together, obtaining an acyclic matching with at most |I nd(G[D])| critical cells. By the weak Morse inequalities, we are done. Definition 9. The decycling number φ(G) of a graph G is the minimum number of vertices whose deletion from G turns it into a forest. Corollary 37. ˜ nd(G))| ≤ |χ(I

X i

dim H˜i (I nd(G); Q) ≤

min |I nd(G[D])| ≤ 2φ(G) . ;6= D⊆V (G),G\D is a forest

Proof. The left-hand side comes from the fact that the Euler characteristic is the alternating some of the ranks of homology groups. The right hand side comes from Definition 9.

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References [1] A. Engström, Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory. [2] J. Jonsson, Simplicial complexes of graphs.

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