A simplicial complex is uniquely determined by its set of discrete Morse functions
AA SIMPLICIAL COMPLEX IS UNIQUELY DETERMINED BY ITS SETOF DISCRETE MORSE FUNCTIONS
NICOLAS ARIEL CAPITELLI AND ELIAS GABRIEL MINIAN
Abstract.
We prove that a connected simplicial complex is uniquely determined by itscomplex of discrete Morse functions. This settles a question raised by Chari and Joswig.In the 1-dimensional case, this implies that the complex of rooted forests of a connectedgraph G completely determines G . Introduction
The complex of discrete Morse functions M ( K ) of a finite simplicial complex K wasintroduced by Chari and Joswig in [5] to study the topology of simplicial complexes interms of their sets of discrete deformations. Despite the potential utility of this complex,very little was known about the relationship between K and M ( K ). Chari and Joswigstudied some properties of the complexes associated to graphs and simplices and computedthe homotopy type of the complex associated to the 2-simplex. Their work was shortlyfollowed by Ayala, Fern´andez, Quintero and Vilches, who described the structure of the pure Morse complex of a graph G , i.e. the subcomplex of M ( G ) generated by the simplicesof maximal dimension [1]. As pointed out in [5], the construction of M ( K ) in the contextof graphs was already implicit in the work of Kozlov [10], who studied complexes arisingfrom directed sub-trees of a given (directed) graph. Kozlov proved shellability of the com-plexes associated to complete graphs and computed the homotopy type of the complexesassociated to paths and cycles.The aim of this article is to settle the connection between a simplicial complex and itscomplex of discrete Morse functions. We show that K is completely determined by M ( K ).Concretely, our main result is the following. Theorem A.
Let
K, L be finite connected simplicial complexes. If M ( K ) is isomorphicto M ( L ) then K is isomorphic to L . For the 1-dimensional case, we prove that Theorem A also holds for multigraphs.
Theorem B.
Let
G, G (cid:48) be finite connected multigraphs. If M ( G ) is isomorphic to M ( G (cid:48) ) then G is isomorphic to G (cid:48) . We also exhibit an example which shows that the homotopy type of M ( K ) does notdetermine the homotopy type of K .The results in this article provide the complete answers to the foundational questionsabout M ( K ) raised by Chari and Joswig in [5].2. The complex of discrete Morse functions
All simplicial complexes that we deal with are assumed to be finite. We write σ ≺ τ ifthe simplex σ is an immediate face of τ (i.e. a proper maximal face) and we let V K denote Mathematics Subject Classification.
Key words and phrases.
Discrete Morse theory, discrete Morse complex, collapsibility.Researchers of CONICET. Partially supported by grants ANPCyT PICT-2011-0812, CONICET PIP112-201101-00746 and UBACyT 20020130100369. a r X i v : . [ m a t h . C O ] S e p N. A. CAPITELLI AND E. G. MINIAN the set of vertices of a complex K . We denote by ∆ n the standard complex consisting ofall the faces of an n -simplex, and by ∂ ∆ n its boundary (i.e. the complex of all the properfaces of the simplex).A discrete Morse funcion f on an abstract simplicial complex K is a map f : K → R satisfying, for every σ ∈ K ,(1) |{ τ (cid:31) σ | f ( τ ) ≤ f ( σ ) }| ≤ |{ ν ≺ σ | f ( ν ) ≥ f ( σ ) }| ≤ | A | denotes the cardinality of the set A . A simplex σ such that both of these numbersare zero is called critical . If f ( η ) ≥ f ( ρ ) for some η ≺ ρ then the pair ( η, ρ ) is called a regular pair . One can easily see that every simplex in K is either critical or belongs to aunique regular pair (see [7, 8] for more details). If ( σ, τ ) is a regular pair, we call σ the source simplex of the pair, and write s ( σ, τ ) = σ , and we call τ the target simplex of thepair, and write t ( σ, τ ) = τ . Typically, a regular pair ( σ, τ ) is depicted graphically as anarrow from σ to τ (see Figure 1). -1 11-2 -3 43 Figure 1.
Graphical representation of regular pairs.
The index of a regular pair ( σ, τ ) is the dimension of σ . A regular pair of index k willbe sometimes denoted by ( σ k , τ k +1 ). Given two discrete Morse functions f, g on K wewrite f (cid:46) g if every regular pair of f is also a regular pair of g . Following [5], if f (cid:46) g and g (cid:46) f (i.e. both functions have the same regular pairs) then we say that they are equivalent . We will make no distinction between equivalent Morse functions, i.e. we willwork with classes of discrete Morse functions under this equivalence relation.A discrete Morse function with exactly one regular pair is called a primitive Morsefunction . We will often identify a primitive Morse function with its sole regular pair. Acollection f , . . . , f r of primitive Morse functions is said to be compatible if there exists adiscrete Morse function f on K with f i (cid:46) f for every i = 0 , . . . , r . The complex of discreteMorse functions of K is the simplicial complex M ( K ) whose vertices are the primitiveMorse functions on K and whose r -simplices are the discrete Morse functions with r + 1regular pairs. We identify in this way a discrete Morse function f with the set { f , . . . , f r } of all primitive Morse functions satisfying f i (cid:46) f (i.e. the set of its regular pairs). M ( K )is also called the discrete Morse complex of K . Figure 2 shows some low-dimensionalexamples of discrete Morse complexes.There is an alternative approach to discrete Morse theory due to Chari [4] where thedeformations are encoded in terms of acyclic matchings in the Hasse diagram of the faceposet of the simplicial complex. It is not hard to see that the pairing of simplices whichform regular pairs of a discrete Morse function determines a matching in the Hasse diagram H K of K . If the arrows in this matching are reversed, it can be easily shown that theresulting directed graph is acyclic. On the other hand, from an acyclic matching on theHasse diagram of a simplicial complex one can build a discrete Morse function f on K where the regular pairs of f are precisely the edges of the matching. From this viewpoint, M ( K ) is the simplicial complex on the edges of the Hasse diagram of K whose simplicesare the subsets of edges which form acyclic matchings. COMPLEX IS DETERMINED BY ITS SET OF DISCRETE MORSE FUNCTIONS 3 ( ) a b a cb(b,e) e (a,e) = a b (c,e ) e (a,e) = c e (b,e)(b,e ) ( )( ) = e ee (a,e) (b,e)(c,e )(b,e ) (c,e ) (a,e ) Figure 2.
Examples of complexes of discrete Morse functions. The complexes associated to graphs
The complex of discrete Morse functions has been studied almost exclusively for graphs,as the construction of M ( K ) for a general K is rather complicated (see for example [1, 5]).We focus first on this case and settle the main result for 1-dimensional regular CW-complexes (Theorem B).Recall that a multigraph G is a triple ( V G , E G , f G ) where V G is a (finite) set of vertices, E G is a set of edges and f G : E G → {{ u, v } : u, v ∈ V G and u (cid:54) = v } is a map whichassigns to each edge its boundary vertices. If f G ( e ) = f G ( e (cid:48) ) for e, e (cid:48) ∈ E G , we say that e, e (cid:48) are parallel edges . For v, v (cid:48) ∈ V G , E G ( v, v (cid:48) ) will stand for the set of parallel edgesbetween v and v (cid:48) . Note that, by definition, a multigraph has no loops. Simple graphscorrespond to multigraphs G where f G is injective. In this case we shall identify an edgewith its boundary vertices and write e = vw if f G ( e ) = { v, w } . Note that simple graphsare precisely the 1-dimensional simplicial complexes and multigraphs are precisely the1-dimensional regular CW-complexes (see [11] for the necessary definitions).The complex of discrete Morse functions of a graph was first studied by Kozlov [10]under a different context. Given a directed graph G , Kozlov defined the simplicial complex∆( G ) whose vertices are the edges of G and whose faces are all directed forests whichare subgraphs of G . In [10] he studied the shellability of the complete double-directedgraph on n vertices (a graph having exactly one edge in each direction between any pairof vertices) and computed the homotopy type of the double-directed n -cycle and thedouble-directed n -path. It is not hard to see that for any (undirected) graph G , theidentity M ( G ) = ∆( d ( G )) holds, where d ( G ) is the directed graph on the vertices of G with one edge in each direction between adjacent vertices of G . The aforementionedexamples studied by Kozlov correspond respectively to the complex of Morse functions ofthe complete graph, the n -cycle and the n -path. Complexes of directed graphs have beenwidely studied (see for example [3, 6, 9, 10]) and some results of this theory were used inBabson and Kozlov’s proof of the Lov´asz conjecture (see [2]).In this section we prove Theorem B, which is the special case of Theorem A for regular1-dimensional CW-complexes. The definition of the complex of Morse functions for regularCW-complexes is identical to the simplicial case. In particular, for a multigraph G , M ( G )can be viewed as the simplicial complex with one vertex for each directed edge in G andwhose simplices are the collections of directed edges which do not form directed cycles.We first establish the result for simple graphs (i.e. the 1-dimensional case of TheoremA) and then extend it to general multigraphs. We begin by collecting some basic factsabout the discrete Morse complex of simple graphs.Given two simplicial complexes K, L , we denote K ≡ L if they are isomorphic. Lemma 3.1.
Let G be a connected simple graph. Then, N. A. CAPITELLI AND E. G. MINIAN (1) | V M ( G ) | = 2 | E G | .(2) dim( M ( G )) = | V G | − .Proof. If G is a tree then it is collapsible and there exists a discrete Morse function f ∈ M ( G ) for which all the edges of G are regular (see [7, Lemma 4.3]). Hence, dim( M ( G )) = | E G | − | V G | −
2. For the general case, proceed by induction on n = | E G | . If G isnot a tree, let f ∈ M ( G ) be of maximal dimension and let e , . . . , e r be a cycle in G .There must be an edge e i which is not regular for f (see [7, Theorem 9.3]). Let G (cid:48) = G − { e i } . G (cid:48) is still connected because e i is in a cycle, | E G (cid:48) | = | E G | − M ( G (cid:48) )) = | V G (cid:48) | − | V G | −
2. Since f ∈ M ( G (cid:48) ) and dim( M ( G (cid:48) )) ≤ dim( M ( G )),then dim( M ( G )) = | V G | − (cid:3) Corollary 3.2. If G, G (cid:48) are connected simple graphs such that M ( G ) ≡ M ( G (cid:48) ) then | V G | = | V G (cid:48) | and | E G | = | E G (cid:48) | . In particular their fundamental groups π ( G ) and π ( G (cid:48) ) are isomorphic. Remark 3.3.
It is easy to check that a vertex v ∈ G is a leaf if and only if the vertex( v, e ) ∈ V M ( G ) is compatible with every other ( u, e (cid:48) ) ∈ V M ( G ) with the unique exception of( w, e ), where w is the other vertex of the edge e . This happens if and only if deg( v, e ) =2 | E G | −
2, where deg( v, e ) is the degree of the vertex ( v, e ) in the 1-skeleton M ( G ) (1) (i.e.the subcomplex of M ( G ) consisting of the simplices of dimension ≤ M ( G ) ≡ M ( G (cid:48) ) then G and G (cid:48) have the same number of leaves.Let C n denote the simple cycle with n vertices. Corollary 3.4.
Let
G, G (cid:48) be two connected simple graphs. If M ( G ) ≡ M ( G (cid:48) ) and G = C n then G (cid:48) = C n .Proof. By a previous result, | V G | = | V G (cid:48) | and | E G | = | E G (cid:48) | . Since G = C n then | V G | = | E G | and therefore | V G (cid:48) | = | E G (cid:48) | . Also, since G has no leaves then G (cid:48) has no leaves. Therefore, G (cid:48) = C n . (cid:3) In order to prove the main results of this paper we will analyze compatibility of regularpairs, similarly as we did in Remark 3.3. From now on, we write ( σ, τ ) ∼ ( η, ρ ) if ( σ, τ ) and( η, ρ ) are compatible as primitive Morse functions (i.e. if they form a simplex in M ( K )),and ( σ, τ ) (cid:28) ( η, ρ ) whenever they are not. Theorem 3.5.
Let
G, G (cid:48) (cid:54) = C n be connected simple graphs and let F : M ( G ) → M ( G (cid:48) ) bea simplicial isomorphism. Define a mapping f : G → G (cid:48) by f ( v ) = s ( F ( v, e )) , where e isany edge incident to v . Then f is a well-defined simplicial isomorphism.Proof. The key part of the proof is to see that f is well-defined, i.e. that f ( v ) does notdepend on the choice of the incident edge e . Suppose otherwise and let ( v, e ) , ( v, e ) ∈ V M ( K ) be such that F ( v, e ) = ( w, a ) and F ( v, e ) = ( w (cid:48) , b ) with w (cid:54) = w (cid:48) . Since ( v, e ) (cid:28) ( v, e ) then ( w, a ) (cid:28) ( w (cid:48) , b ) and hence a = b (see Figure 3). v e e w w ´ F a b= Figure 3
We claim that under this situation we can choose such a vertex v of G with degreegreater than or equal to 3. This will lead to a contradiction since an edge containing v different from e and e provides a primitive Morse function on G which is incompatiblewith both ( v, e ) and ( v, e ), while the simplicity of G (cid:48) implies that there is no possible COMPLEX IS DETERMINED BY ITS SET OF DISCRETE MORSE FUNCTIONS 5 primitive Morse function on G (cid:48) incompatible with both ( w, a ) and ( w (cid:48) , a ). To prove thisclaim, let e = vv (cid:48) and consider the primitive Morse function ( v (cid:48) , e ). Since ( w (cid:48) , a ) = F ( v, e ) (cid:28) F ( v (cid:48) , e ) and F is an isomorphism then there exists and edge c = w (cid:48) w (cid:48)(cid:48) ∈ G (cid:48) such that F ( v (cid:48) , e ) = ( w (cid:48) , c ). Consider now ( w (cid:48)(cid:48) , c ) ∈ M ( G (cid:48) ). Using a similar argumentfor F − and ( w (cid:48)(cid:48) , c ) one can find an edge e (cid:54) = e , e such that F − ( w (cid:48)(cid:48) , c ) = ( v (cid:48) , e ) (seeFigure 4). v e e w w ´ F a b=v ´ c w ´´ e Figure 4
Note that the primitive Morse functions ( v (cid:48) , e ) , ( v (cid:48) , e ) satisfy the same hypotheses than( v, e ) , ( v, e ) (but replacing ( w, a ) , ( w (cid:48) , a ) with ( w (cid:48) , c ) , ( w (cid:48)(cid:48) , c ) respectively). Repeating thisargument we obtain a path e , e , e , . . . where, for any vertex v ∈ e i ∩ e i +1 , ( v, e i ) , ( v, e i +1 )are mapped to primitive Morse functions on G (cid:48) of the form ( u, d ) , ( u (cid:48) , d ) with u (cid:54) = u (cid:48) . Byfiniteness, this path must form a cycle C = { e j , e j +1 , . . . , e j + k − , e j + k = e j } for some j, k .If j = 0, and since G is not a cycle, there is by connectedness an edge e / ∈ C intersecting C . In this case, x = e ∩ C is the desired vertex (see Figure 5 ( a )). If j > y = e j − ∩ e j is the desired vertex (see Figure 5 ( b )). This proves that f is well-defined. e e e e k-1 e k e x e j e j+1 e j+2 e j-1 e j+k-1 y e j-2 e j+k-2 a ( ) b ( ) Figure 5
We show now that f is a simplicial morphism. Consider an edge e = vv (cid:48) ∈ G . Wemust see that f ( v ) f ( v (cid:48) ) ∈ G (cid:48) . Since ( v, e ) (cid:28) ( v (cid:48) , e ) then F ( v, e ) (cid:28) F ( v (cid:48) , e ). Therefore,either s ( F ( v, e )) = s ( F ( v (cid:48) , e )) or t ( F ( v, e )) = t ( F ( v (cid:48) , e )). In the first case, the samereasoning as above applied to h = s ◦ F − : G (cid:48) → G gives a contradiction. Therefore, t ( F ( v, e )) = t ( F ( v (cid:48) , e )) and, in particular, f ( v ) f ( v (cid:48) ) ∈ t ( F ( v, e )) is an edge in G (cid:48) .Finally, it is easy to see that f − = s ◦ F − is the inverse of f . (cid:3) Corollary 3.6.
Let
G, G (cid:48) be connected simple graphs. If M ( G ) ≡ M ( G (cid:48) ) then G ≡ G (cid:48) .Proof. Follows from Corollary 3.4 and Theorem 3.5. (cid:3)
We now extend the result to multigraphs. Two primitive Morse functions ( v, e ) , ( v (cid:48) , e (cid:48) ) ∈ M ( G ) are said to be parallel if v = v (cid:48) and e is parallel to e (cid:48) in G . Recall that the link ofa simplex σ ∈ K is the subcomplex lk ( σ, K ) = { τ ∈ K : τ ∩ σ = ∅ , τ ∪ σ ∈ K } . Lemma 3.7.
Let G be a connected multigraph with more than two vertices. Then twoprimitive Morse functions ( v, e ) , ( v (cid:48) , e (cid:48) ) are parallel in M ( G ) if and only if ( v, e ) (cid:28) ( v (cid:48) , e (cid:48) ) and lk (( v, e ) , M ( G )) = lk (( v (cid:48) , e (cid:48) ) , M ( G )) .Proof. Suppose first that ( v, e ) (cid:28) ( v (cid:48) , e (cid:48) ) and lk (( v, e ) , M ( G )) = lk (( v (cid:48) , e (cid:48) ) , M ( G )). If ( v, e )and ( v (cid:48) , e (cid:48) ) are not parallel in M ( G ), then there are only three possibilities for the edges e and e (cid:48) in G which are shown in Figure 6. N. A. CAPITELLI AND E. G. MINIAN a ( ) b ( ) c ( ) v ve e v v=e e v ve e= Figure 6
Since | V G | ≥ G is connected, in each of the three cases, G locally looks as inFigure 7. a ( ) b ( ) c ( ) v ve e v v= e v ve e=w e w Figure 7
This contradicts the fact that lk (( v, e ) , M ( G )) = lk (( v (cid:48) , e (cid:48) ) , M ( G )). The other implica-tion is trivial. (cid:3) Given a simplicial complex K , we define an equivalence relation R on V K as follows: v R w ⇔ v = w or { v, w } / ∈ K and lk ( v, K ) = lk ( w, K ) . Let (cid:101) K be the simplicial complex whose vertices are the equivalence classes of vertices of K and whose simplices are the sets { ˜ v , . . . , ˜ v r } such that { v , . . . , v r } ∈ K . Here ˜ v denotesthe equivalence class of the vertex v . Note that (cid:101) K is well-defined since, if v i R v (cid:48) i then { v , . . . , v i , . . . , v r } ∈ K if and only if { v , . . . , v (cid:48) i , . . . , v r } ∈ K . Proposition 3.8.
Let
K, L be simplicial complexes and let ˜ K and ˜ L be as above. If f : K → L is a simplicial isomorphism then the map ˜ f : (cid:101) K → (cid:101) L given by ˜ f (˜ v ) = (cid:103) f ( v ) isa simplicial isomorphism.Proof. We prove first that ˜ f is well-defined. Suppose v R v (cid:48) with v (cid:54) = v (cid:48) . Since { v, v (cid:48) } / ∈ K and f is an isomorphism then { f ( v ) , f ( v (cid:48) ) } / ∈ L . Also, if { f ( v ) }∪ σ ∈ L then { v }∪ f − ( σ ) ∈ K , which implies that { v (cid:48) } ∪ f − ( σ ) ∈ K . Therefore f ( v (cid:48) ) ∪ σ ∈ L .Finally, ˜ f is an isomorphism since ˜ f − = (cid:103) f − . (cid:3) Definition.
For a multigraph G we define the simplification of G , denoted by sG , as thesimple graph obtained from G by identifying parallel edges. Remark 3.9.
By Lemma 3.7 one can check that the map f : (cid:94) M ( G ) → M ( sG ) defined by f ( (cid:93) ( v, e )) = ( v, e ) is a well-defined isomorphism. Here e is the image of the edge e in sG . Proof of Theorem B.
Let F : M ( G ) → M ( G (cid:48) ) be an isomorphism. By Proposition 3.8 andRemark 3.9, F induces an isomorphism M ( sG ) → M ( sG (cid:48) ) which we also denote by F .By Theorem 3.5 there is an isomorphism f : sG → sG (cid:48) sending a vertex v to s ( F ( v, e ))for any edge e incident to v . Then, in order to see that G and G (cid:48) are isomorphic, we onlyneed to check that | E G ( v, w ) | = | E G (cid:48) ( f ( v ) , f ( w )) | for any pair of vertices v, w of G .We can suppose that | E G ( v, w ) | (cid:54) = 0 and choose some e ∈ E G ( v, w ). Then ( v, e ) ∈ M ( G )and let e (cid:48) = t ( F ( v, e )) ∈ E G (cid:48) ( f ( v ) , f ( w )). Note that the set E G ( v, w ) is in bijection withthe set { ( v, a ) ∈ M ( G ) , ( v, a ) (cid:28) ( w, e ) } . Similarly, E G (cid:48) ( f ( v ) , f ( w )) is in bijection with { ( f ( v ) , a (cid:48) ) ∈ M ( G (cid:48) ) , ( f ( v ) , a (cid:48) ) (cid:28) ( f ( w ) , e (cid:48) ) } . By the isomorphism F , both sets have thesame cardinality. (cid:3) COMPLEX IS DETERMINED BY ITS SET OF DISCRETE MORSE FUNCTIONS 7
Chari and Joswig asked in [5] whether there is any connection between the homotopytypes of K and M ( K ). They implicitly showed that the homotopy type of K does notdetermine the homotopy type of M ( K ). For instance, by [5, Proposition 5.1] the complexof Morse functions associated to the 1-simplex is homotopy equivalent to S and the oneassociated to the 2-simplex is homotopy equivalent to S ∨ S ∨ S ∨ S . The followingexample shows that the homotopy type of M ( K ) does not determine the homotopy typeof K either. Example 3.10.
Consider the following simple graphs. G has three vertices u, v, w andtwo edges uv, uw . The graph G (cid:48) has four vertices a, b, c, d and four edges ab, bc, ac, ad .Note that they are not homotopy equivalent while their associated complexes of Morsefunctions are both contractible.4. Proof of the main result
We now extend the result of Corollary 3.6 to simplicial complexes of any dimension.The idea behind the proof is that, in “almost all” cases, a simplicial isomorphism F : M ( K ) → M ( L ) restricts to an isomorphism F | M ( K (1) ) : M ( K (1) ) → M ( L (1) ) between the1-skeletons and by Theorem 3.5 the 1-skeletons of K and L are isomorphic. Then aninductive argument shows that an isomorphism M ( K ) ≡ M ( L ) forces all skeletons of K and L to be isomorphic.In the following we will use Forman’s concept of V -path associated to a discrete vectorfield V over a complex K . Given a discrete Morse function f : K → R , an f -path ofindex k is a sequence of regular k -simplices σ , . . . , σ r ∈ K such that σ i (cid:54) = σ i +1 for all0 ≤ i ≤ r − σ i +1 ≺ τ i , where τ i is the target of the regular pair with source σ i .This is actually the notion of a V f -path , where V f is the discrete gradient vector field of f . The f -path is called closed if σ = σ r and non-stationary if σ (cid:54) = σ . We shall beexclusively dealing with non-stationary closed f -paths, so we will simply refer to them as f -cycles . Note that an f -cycle of index k is equivalent to having an incompatible collection P = { ( σ , τ ) , . . . , ( σ r , τ r ) } of primitive Morse functions of index k ≥ P is compatible. Equivalently, the full subcomplex of M ( K ) spanned bythe vertices ( σ , τ ) , . . . , ( σ r , τ r ) is the boundary ∂ ∆ r of an r -simplex.Note that an f -cycle has at least three primitive Morse functions. One with exactlythree primitive Morse functions is said to be minimal and two minimal f -cycles sharingexactly one regular pair are said to be adjacent . From the mutually exclusive natureof properties (1) and (2) in page 2 we see that no collection of regular pairs of a givencombinatorial Morse function admits f -cycles of any index. Actually, Forman provedthat this property characterizes the discrete vector fields that arise from a discrete Morsefunction (see [7, Theorem 9.3]). Remarks 4.1. ( i ) Note that a cycle e , . . . , e r in the 1-skeleton of a complex K gives rise to twopossible f -cycles of index 0 in K : choosing a vertex v for e , one of them is { ( v , e ) , ( v , e ) , . . . , ( v r , e r ) } where v i (cid:54) = v i +1 for all i = 0 , . . . , r −
1. The other f -cycle arises from selecting the other vertex of e to be the source of the primitiveMorse function.( ii ) It is easy to see that if { ( σ , τ ) , ( σ , τ ) , ( σ , τ ) } is a minimal f -cycle of index k − { τ , τ , τ } spans a complex with k + 2 vertices and a complete 1-skeleton.The following result deals with the cases in which an isomorphism M ( K ) → M ( L ) doesnot restrict to an isomorphism M ( K (1) ) → M ( L (1) ). N. A. CAPITELLI AND E. G. MINIAN
Proposition 4.2.
Let
K, L be connected simplicial complexes and let F : M ( K ) → M ( L ) be a simplicial isomorphism. If there exists a primitive Morse function ( v, e ) ∈ V M ( K ) ofindex such that F ( v, e ) = ( σ n − , τ n ) with n ≥ , then K = L = ∂ ∆ m for some m ≥ .Proof. We may assume that n is maximal with the property that there exists ( v, e ) ∈ V M ( K ) of index 0 whose image is ( σ n − , τ n ) for some n ≥
2. With this assumption, weshall prove that K = ∂ ∆ n +1 . Let w be the other end of e and consider F ( w, e ). Since n ≥ F ( v, e ) and F ( w, e ) at the same time, thus e must be a face of a 2-simplex { v, w, u } ∈ K . Let e (cid:48) = wu and e (cid:48)(cid:48) = uv and consider the minimal f -cycle { ( v, e ) , ( w, e (cid:48) ) , ( u, e (cid:48)(cid:48) ) } in K . Then { F ( v, e ) , F ( w, e (cid:48) ) , F ( u, e (cid:48)(cid:48) ) } is a minimal f -cycle of index n − L . Let F ( v, e ) = ( σ, τ ), F ( w, e (cid:48) ) = ( σ (cid:48) , τ (cid:48) ) and F ( u, e (cid:48)(cid:48) ) = ( σ (cid:48)(cid:48) , τ (cid:48)(cid:48) ). A simple reasoning shows that if σ (cid:48) ≺ τ thenthe situation of Figure 8 would arise, which leads to a contradiction. v wu ee e (cid:115) (cid:116) F (cid:115)(cid:115) (cid:116)(cid:116) F (v,e ) F (w,e) F (u,e ) (cid:98)(cid:97) (cid:100) F ( ) -1 (cid:97) F ( ) -1 (cid:98) F ( ) -1 (cid:100) Figure 8.
The image of ( v, e (cid:48)(cid:48) ), ( u, e (cid:48) ) and ( w, e ) in the case σ (cid:48) ≺ τ . If we consider a minimal f -cycle of index n − { α, β, δ } in τ (in white arrows) then its preimage by F does not constitutean f -cycle in K , which contradicts the fact that F is an isomorphism. Therefore, we must have σ ≺ τ (cid:48)(cid:48) and the situation is as shown in Figure 9. Let Q bethe subcomplex generated by the n -simplices τ, τ (cid:48) , τ (cid:48)(cid:48) and note that Q has n + 2 ≥ ii )). Let S denote the collection ofall primitive Morse function in Q of index 0 and let G ( x, a ) = t ( F − ( x, a )) ∈ K for each( x, a ) ∈ S . We will prove that K = ∂ ∆ n +1 in various steps. (cid:115) (cid:116) F (cid:115)(cid:115) (cid:116)(cid:116) F (v,e ) F (w,e) F (u,e ) v wu ee e Figure 9
Step 1.
We show first that G ( S ) is a collection of k -simplices for a fixed k ≤ n .Consider a sequence τ = η n (cid:31) σ = η n − (cid:31) η n − (cid:31) · · · (cid:31) η (cid:31) η = y of faces of the n -simplex τ ending in a vertex y of τ . Each pair ( η i − , η i ) is incompatible with the previousand the next pair. Since incompatibility for a given regular pair only happens with regularpairs of one dimension up, one dimension down or of the same dimension, we concludethat F − ( y, η ) = ( ψ k − , ρ k ) for some k ≤ n . Now, since Q has a complete 1-skeleton thenany edge a ∈ Q is part of a cycle also containing η . Therefore, any ( x, a ) ∈ S is part ofan f -cycle of index 0 containing either ( y, η ) or ( z, η ), where z is the other end of η (seeRemark 4.1 ( i )). Since by definition F maps f -cycles to f -cycles, it suffices to show that t ( F − ( z, η )) is also a k -simplex. But since | V Q | ≥
4, we can form an f -cycle of index 0containing ( y, η ) and a new pair ( p, ψ ), and another one containing ( z, η ) and ( p, ψ ) asshown in Figure 10. COMPLEX IS DETERMINED BY ITS SET OF DISCRETE MORSE FUNCTIONS 9 (y, (cid:32) (cid:32) ) (p, ) (p, )(z, ) (cid:121) (cid:104) (cid:104) (cid:121) y yz zp p Figure 10
Step 2.
We show that k = n and that G ( S ) spans ∂ ∆ n +1 . Fix a minimal f -cycle C = { ( v , v v ) , ( v , v v ) , ( v , v v ) } in Q and let T be the subcomplex of K generatedby the three k -simplices in G ( C ). Note that | V T | = k + 2 by Remark 4.1 ( ii ). Weclaim that all k -simplices in G ( S ) have their vertices in V T . To see this, let ( x, a ) ∈ S and let y be the other end of a . All possible situations for ( x, a ) with respect to C arecontemplated in Figure 11 where one can verify that it is always possible to find a sequenceof adjacent minimal f -cycles between C and a minimal f -cycle containing ( x, a ). By an v v v x y (x,a) v v v xy = (x,a) v v v yx = (x,a) v wv x x= (x,a) v y =
123 678 59 4 a ( ) b ( ) c ( ) d ( ) Figure 11.
The sequence of adjacent minimal f -cycles in situation ( d ) is given by C = { , , } , { , , } , { , , } and { , , x, a ) } . inductive argument it suffices to show that the image by G of a regular pair in a minimal f -cycle adjacent to C has it vertices in V T . Let C = { ( v , v v ) , ( v , v v ) , ( v , v v ) } be ageneric minimal f -cycle adjacent to C . Since the k -simplex G ( v , v v ) ∈ G ( C ) ∩ G ( C ),by Remark 4.1 ( ii ) it suffices to show that the only vertex q ∈ V T \ V G ( v ,v v ) is also in G ( C ). But since ( v , v v ) (cid:28) ( v , v v ) then either s ( F − ( v , v v )) = s ( F − ( v , v v )) or t ( F − ( v , v v )) = t ( F − ( v , v v )). The situation must be as shown in Figure 9 and thepossible cases are shown in Figure 12. This proves that q ∈ G ( C ).Now, since Q has a complete 1-skeleton then we can form a cycle in Q (1) containing allthe vertices of Q . The corresponding f -cycle of index 0 has as a preimage by F an f -cycleof index k − n + 2 regular pairs. By definition, the target of all these pairs aredistinct k -simplices. Therefore, we conclude that k = n and that G ( S ) spans ∂ ∆ n +1 . Step 3.
We show that K is spanned by G ( S ). First, note that two primitive Morsefunctions ( x, a ) , ( x, b ) ∈ S of index 0 sharing the same source vertex x ∈ V Q are mappedby F − to primitive Morse functions with the same target n -simplex (i.e. G ( x, a ) = G ( x, b )). To see this, note that since F − ( x, a ) (cid:28) F − ( x, b ) then, by Step 1 , either s ( F − ( x, a )) = s ( F − ( x, b )) or G ( x, a ) = G ( x, b ). Assume the first case holds and let( x, c ) ∈ S with c (cid:54) = a, b . Note that such a pair ( x, c ) exists because n ≥
2. Since the only
G( , )v v v F ( , )v v v s(F ( , ))=v v v s(F ( , ))v v v F ( , )v v v F ( , )v v v q F ( , )v v v t(F ( , ))=v v v t(F ( , ))v v v F ( , )v v v q F ( , )v v v F ( , )v v v F ( , )v v v F ( , )v v v G( , )v v v F ( , )v v v Figure 12.
Here F − ( v , v v ) is drawn with a dashed arrow. On the left: the case s ( F − ( v , v v )) = s ( F − ( v , v v )) cannot happen because we get more than k + 2 vertices.On the right: in the case t ( F − ( v , v v )) = t ( F − ( v , v v )) we readily see that q ∈ G ( C ). primitive Morse functions incompatible with both F − ( x, a ) and F − ( x, b ) have source s ( F − ( x, a )) = s ( F − ( x, b )), there exists an n -simplex in G ( S ) with one vertex q not in V G ( x,a ) ∪ V G ( x,b ) (see Figure 13). x a F -1 bc F (x,a) -1 F (x,b) -1 F (x,c) -1 q Figure 13
This is a contradiction since, by the reasoning made in
Step 2 , all the vertices in G ( S )are included in the set of n + 2 vertices determined by any two distinct n -simplices in G ( S ). We conclude that G ( x, a ) = G ( x, b ), thus we have a bijection between V Q and G ( S ). Suppose now that K − (cid:104) G ( S ) (cid:105) (cid:54) = ∅ . Here (cid:104) G ( S ) (cid:105) denotes the subcomplex spannedby G ( S ). Let ˜ e ∈ K be an edge such that ˜ e ∩ (cid:104) G ( S ) (cid:105) consists of a vertex z . Considerthe primitive Morse function ( z, ˜ e ) and let ( z, ˜ e (cid:48) ) , ( z, ˜ e (cid:48)(cid:48) ) ∈ (cid:104) G ( S ) (cid:105) . Since ( z, ˜ e ) (cid:28) ( z, ˜ e (cid:48) )and ( z, ˜ e ) (cid:28) ( z, ˜ e (cid:48)(cid:48) ) then F ( z, ˜ e ) (cid:28) F ( z, ˜ e (cid:48) ) and F ( z, ˜ e ) (cid:28) F ( z, ˜ e (cid:48)(cid:48) ). Since n is maximaland t ( F ( z, ˜ e (cid:48) )) = t ( F ( z, ˜ e (cid:48)(cid:48) )) by the previous reasoning, then t ( F ( z, ˜ e )) must be equal to t ( F ( z, ˜ e (cid:48) )) = t ( F ( z, ˜ e (cid:48)(cid:48) )). This is a contradiction because, due to the bijection between V Q and G ( S ), all n + 1 regular pairs whose target is this n -simplex are in the image of the n + 1 regular pairs in S with source z . This concludes the proof. (cid:3) Similarly as we did with the edges of simple graphs, for simplicity of notation, an n -simplex σ = { v , . . . , v n } ∈ K will be denoted by σ = v · · · v n . Proof of Theorem A.
Let F : M ( K ) → M ( L ) be an isomorphism. By Proposition 4.2 wemay assume that every primitive Morse function of index 0 in M ( K ) (resp. in M ( L ))is mapped by F (resp. by F − ) to a primitive Morse function of index 0. This gives awell-defined isomorphism F | M ( K (1) ) : M ( K (1) ) → M ( L (1) ). By Theorem 3.5 there existsan isomorphism f : K (1) → L (1) with f ( v ) = s ( F ( v, e )) for any e (cid:31) v . Note that for everyedge xy ∈ K we have F ( x, xy ) = ( f ( x ) , f ( x ) f ( y )). We will show by induction that forany ( n + 1)-simplex v · · · v n +1 , F ( v · · · v n , v · · · v n v n +1 ) = ( f ( v ) · · · f ( v n ) , f ( v ) · · · f ( v n ) f ( v n +1 )) . Given τ = v · · · v n +1 ∈ K , consider the following two families of primitive Morsefunctions: COMPLEX IS DETERMINED BY ITS SET OF DISCRETE MORSE FUNCTIONS 11 • I = { ( v · · · ˆ v i · · · v n , v · · · v n ) , ≤ i ≤ n }• J = { ( v · · · ˆ v j · · · v n +1 , v · · · v n +1 ) , ≤ j ≤ n + 1 } ,where the hat over a vertex means that that vertex is to be omitted. By induction, • F ( v · · · ˆ v i · · · v n , v · · · v n ) = ( f ( v ) · · · (cid:91) f ( v i ) · · · f ( v n ) , f ( v ) · · · f ( v n )) and • F ( v · · · ˆ v j · · · v n +1 , v · · · v n +1 ) = ( f ( v ) · · · (cid:91) f ( v j ) · · · f ( v n +1 ) , f ( v ) · · · f ( v n +1 )). Since ( v · · · v n , v · · · v n +1 ) ∈ M ( K ) is incompatible with every element of I then F ( v · · · v n , v · · · v n +1 ) = ( f ( v ) · · · f ( v n ) , f ( v ) · · · f ( v n ) w )for some vertex w ∈ L . On the other hand, since ( v · · · v n +1 , v · · · v n +1 ) ∈ M ( K ) isincompatible with every element of J then F ( v · · · v n +1 , v · · · v n +1 ) = ( f ( v ) · · · f ( v n +1 ) , f ( v ) · · · f ( v n +1 ) u )for some vertex u ∈ L . But ( v · · · v n , v · · · v n +1 ) (cid:28) ( v · · · v n +1 , v · · · v n +1 ), so we musthave f ( v ) · · · f ( v n ) w = f ( v ) · · · f ( v n +1 ) u , and therefore w = f ( v n +1 ) and u = f ( v ). (cid:3) References [1] R. Ayala, L. M. Fern´andez, A. Quintero, and J. A. Vilches.
A note on the pure Morse complex of agraph.
Topology Appl. 155 (2008), No. 17-18, 2084-2089.[2] E. Babson, D. N. Kozlov.
Proof of the Lov´asz conjecture.
Ann. of Math. (2) 165 (2007), No. 3, 965-1007.[3] A. Bj¨orner, V. Welker.
Complexes of directed graphs.
SIAM J. Discrete Math. 12 (1999), No. 4, 413-424(electronic).[4] M. K. Chari.
On discrete Morse functions and combinatorial decompositions . Discrete Math. 217(2000), No. 1-3, 101-113.[5] M. K. Chari, M. Joswig.
Complexes of discrete Morse functions.
Discrete Math. 302 (2005), No. 1-3,3951.[6] A. Engstr¨om.
Complexes of directed trees and independence complexes.
Discrete Math. 309 (2009),No. 10, 3299-3309.[7] R. Forman.
Morse theory for cell complexes.
Adv. Math. 134 (1998), No. 1, 90-145.[8] R. Forman.
Witten-Morse theory for cell complexes.
Topology 37 (1998), No. 5, 945-979.[9] D. Joji´c.
Shellability of complexes of directed trees.
Filomat 27 (2013), No. 8, 1551-1559.[10] D. N. Kozlov.
Complexes of directed trees.
J. Combin. Theory Ser. A 88 (1999), No. 1, 112-122.[11] A. Lundell, S. Weingram.
The topology of CW complexes.
Van Nostrand - Reinhold, New York (1969).
Departamento de Matem´atica-IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires,Argentina.
E-mail address : [email protected] E-mail address ::