On the gap between representability and collapsibility
aa r X i v : . [ m a t h . C O ] M a r On the gap between representability and collapsibility
Jiˇr´ı Matouˇsek Martin Tancer
Department of Applied Mathematics andInstitute of Theoretical Computer Science (ITI)Charles University, Malostransk´e n´am. 25118 00 Praha 1, Czech Republic
November 1, 2018
Abstract
A simplicial complex K is called d -representable if it is the nerve of acollection of convex sets in R d ; K is d -collapsible if it can be reduced toan empty complex by repeatedly removing a face of dimension at most d − K is d -Leray ifevery induced subcomplex of K has vanishing homology of dimension d and larger.It is known that d -representable implies d -collapsible implies d -Leray,and no two of these notions coincide for d ≥
2. The famous Helly the-orem and other important results in discrete geometry can be regardedas results about d -representable complexes, and in many of these results“ d -representable” in the assumption can be replaced by “ d -collapsible” oreven “ d -Leray”.We investigate “dimension gaps” among these notions, and we con-struct, for all d ≥
1, a 2 d -Leray complex that is not (3 d − d -collapsible complex that is not (2 d − d , is d -collapsible. (ii) If the nerve of asimplicial complex K is d -representable, then K embeds in R d . d -representability. Helly’s theorem [Hel23] asserts that if C , C , . . . , C n areconvex sets in R d , n ≥ d + 1 and every d + 1 of the C i have a common point,then T ni =1 C i = ∅ . This famous theorem and many others in discrete geometrydeal with intersection patterns of convex sets in R d , and they can be restatedusing the notion of d -representable simplicial complexes.We recall that the nerve N ( S ) of a family S = { S , S , . . . , S n } is the simpli-cial complex with vertex set [ n ] := { , , . . . , n } and with a set σ ⊆ [ n ] forminga simplex if T i ∈ σ S i = ∅ . A simplicial complex K is d -representable if it is iso-morphic to the nerve of a family of convex sets in R d (all simplicial complexesthroughout this paper are assumed to be finite).In this language Helly’s theorem implies that a d -representable complex isdetermined by its d -skeleton. Other examples of theorems that can be seen as1 σ = { , }
12 34 σ = { , }
12 34 σ = { } σ = { , } σ = { } σ = ∅ ( V, ∅ ) Figure 1: An example of 2-collapsing.statements about d -representable complexes include the fractional Helly theo-rem of Katchalski and Liu [KL79], the colorful Helly theorem of Lov´asz ([Lov74];also see [B´ar82]), the ( p, q )-theorem of Alon and Kleitman [AK92], and theHelly-type result of Amenta [Ame96] (conjecture by Gr¨unbaum and Motzkin).Among the deepest results concerning d -representable complexes is a completecharacterization of their f -vectors conjectured by Eckhoff and proved by Kalai[Kal86, Kal84]. We also refer to [DGK63, Eck93, Mat02] for more examplesand background. d -collapsibility and d -Leray complexes. Wegner in his seminal 1975 paper[Weg75] introduced d -collapsible simplicial complexes. To define this notion, wefirst introduce an elementary d -collapse . Let K be a simplicial complex and let σ, τ ∈ K be faces (simplices) such that(i) dim σ ≤ d − τ is an inclusion-maximal face of K ,(iii) σ ⊆ τ , and(iv) τ is the only face of K satisfying (ii) and (iii).Then we say that σ is a d -collapsible face of K and that the simplicial complex K ′ := K \ { η ∈ K : σ ⊆ η ⊆ τ } arises from K by an elementary d -collapse.A simplicial complex K is d -collapsible if there exists a sequence of elementary d -collapses that reduces K to the empty complex ∅ . Fig. 1 shows an example of2-collapsing.Another related notion is a d -Leray simplicial complex, where K is d -Leray ifevery induced subcomplex of K (i.e., a subcomplex of the form K [ X ] := { σ ∩ X : σ ∈ K } for some subset X of the vertex set V ( K )) has zero homology (over Q )in dimension d and larger.Wegner [Weg75] proved that every d -representable complex is d -collapsibleand every d -collapsible complex is d -Leray. By inspecting proofs of several theo-rems about intersection patterns of convex sets in R d , i.e. about d -representablecomplexes, one can sometimes see that they actually use only d -collapsibility,and thus they are valid for all d -collapsible complexes (good examples, amongthose mentioned earlier, are the fractional Helly theorem and the colorful Hellytheorem).With more work it has been shown that all of the results mentioned aboveand some others also hold for d -Leray complexes. For example, for Helly’s theo-rem this follows essentially from Helly’s own topological generalization [Hel30],for the ( p, q )-theorem this was proved in [AKMM02], and for the colorful Helly The f -vector of a d -dimensional simplicial complex K is the integer vector ( f , f , . . . , f d ),where f i is the number of i -dimensional simplices in K . Figure 3: The triangulation of the dunce hat [Weg75]; vertices with the samenumbers should be identified.theorem and for Amenta’s theorem this was shown recently by Kalai and Meshu-lam [KM05, KM07]. Kalai’s characterization of f -vectors of d -representablecomplexes is also valid for the f -vectors of d -Leray complexes, showing that f -vectors cannot distinguish these classes.These results indicate that the notions of d -representable, d -collapsible,and d -Leray are similar in some important respects. However, no two ofthem coincide. Fig. 2 shows an example of a 1-collapsible complex that isnot 1-representable. Wegner [Weg75] noted that well-known examples of 2-dimensional complexes that are contractible but not collapsible, such as suitabletriangulations of the “dunce hat” (Fig. 3) or Bing’s house (see, e.g., [Hat01]),are 2-Leray but not 2-collapsible. Results.
The goal of the present paper is to exhibit stronger differencesamong these notions; more precisely, to investigate “dimension gaps”. We set ρ ( K ) := min { d : K is d -representable } (“representability”) ,γ ( K ) := min { d : K is d -collapsible } (“collapsibility”) ,λ ( K ) := min { d : K is d -Leray } (“Leray number”) . Theorem 1.1. (a)
For every d ≥ there exists a complex K with γ ( K ) = d and ρ ( K ) =2 d − (i.e., d -collapsible and not (2 d − -representable). (b) For every d ≥ there exists a complex K with λ ( K ) = 2 d and γ ( K ) = 3 d (i.e., d -Leray and not (3 d − -collapsible). In part (a), our example is the nerve of a d -dimensional simplicial complex L that is not embeddable in R d − . A well known example of such L is the3 -skeleton of the (2 d + 2)-dimensional simplex, due to Van Kampen [vK32] andFlores [Flo34]. The proof of Theorem 1.1(a) then follows immediately from thetwo propositions below, which may be of independent interest. Proposition 1.2.
Let L be a simplicial complex such that the nerve N ( L ) is d -representable. Then L embeds in R d , even linearly. Proposition 1.3.
Let F be a finite family of sets, each of size at most d . Thenthe nerve N ( F ) is d -collapsible. For part (b) of Theorem 1.1, our example is a d -fold join the dunce hattriangulation from Fig. 3. Open problems.
The main questions, which we unfortunately haven’t solved,are: Can representability be bounded in terms of collapsibility (formally, isthere a function f such that ρ ( K ) ≤ f ( γ ( K )) for all K )? Can collapsibilitybe bounded in terms of the Leray number (formally, is there a function f such that γ ( K ) ≤ f ( λ ( K )) for all K )? Theorem 1.1 shows f ( d ) ≥ d − f (2 d ) ≥ d − f thanTheorem 1.1(a), since every d -dimensional complex embeds in R d − . A 2-collapsible complex whose representability might perhaps be unbounded wasnoted by Alon et al. [AKMM02], namely, a finite projective plane (regarded asa simplicial complex, where the lines of the projective plane are the maximalsimplices). More generally, any almost-disjoint set system is easily seen to be2-collapsible, and it would be interesting to decide whether all almost-disjointsystems are d -representable for some constant d . In this section we will prove Proposition 1.2. First we recall a classical lemmaof Radon ([Rad21]; also see, e.g., [Eck93] or [Mat02]) in the following form:
Lemma 2.1.
Let P be a set of affinely dependent points in R d . Then there existtwo disjoint affinely independent subsets A, B ⊂ P with conv( A ) ∩ conv( B ) = ∅ . We will also need the following result of a similar flavor:
Lemma 2.2.
Let A and B be finite subsets of R d . Suppose that there is apoint x ∈ (conv( A ) ∩ conv( B )) \ conv( A ∩ B ) . Then there exist disjoint affinelyindependent sets A ′ ⊆ A and B ′ ⊆ B such that conv( A ′ ) ∩ conv( B ′ ) = ∅ .Proof. The proof is similar to the usual proof of Radon’s lemma, only slightlymore complicated.We can write x as a convex combination of points of A : x = X a ∈ A α a a, (1)where α a ≥ a ∈ A and P a ∈ A α a = 1. Similarly x = X b ∈ B β b b (2)4here β b ≥ b ∈ B and P b ∈ B β b = 1. Let K := A ∩ B and let K + := { p ∈ K : α p > β p } , K − := K \ K + . We define the sets A := A \ K − and B := B \ K + , and we note that A ∩ B = ∅ and A ∪ B = A ∪ B . We claim that conv( A ) ∩ conv( B ) = ∅ ; this will implythe lemma, since the desired affinely independent A ′ and B ′ can be obtainedfrom A and B by removing redundant points.For notational convenience we extend the definition of α p and β p to all p ∈ A ∪ B by letting α p = 0 for p A and β p = 0 for p B . By subtracting(2) from (1) and rearranging we get X p ∈ A ( α p − β p ) p = X p ∈ B ( β p − α p ) p. All coefficients on both sides of this equation are nonnegative. Let us set S := P p ∈ A ( α p − β p ). Since P p ∈ A α p = P p ∈ B β p = 1, we also have S = P p ∈ B ( β p − α p ). Moreover, since x conv( K ), at least one α p with p ∈ A \ K is nonzero,and thus S = 0. We set y := 1 S X p ∈ A ( α p − β p ) p = 1 S X p ∈ B ( β p − α p ) p ;thus, y is expressed as a convex combination of points of A and also as a convexcombination of points of B . Hence conv( A ) ∩ conv( B ) = ∅ as claimed. Proof of Proposition 1.2.
Let L be a simplicial complex such that N ( L ) is d -representable. This means that there exists a system ( C σ : σ ∈ L ) of convex setsin R d such that for every collection M ⊆ L of simplices we have T σ ∈M C σ = ∅ iff T M = ∅ .For every v ∈ V ( L ) we fix a point p ( v ) ∈ T τ ∈ L : v ∈ τ C τ (this intersection isnonempty since v ∈ T { τ ∈ L : v ∈ τ } ).This defines a mapping p : V ( L ) → R d . For every σ ∈ L we set D σ := conv( p ( σ )) . We claim that each D σ is a simplex in R d and that the D σ form a geometricrepresentation of K in R d . To this end, it suffices to verify that the set p ( σ ) isaffinely independent for every σ ∈ L , and that D σ ∩ D τ = D σ ∩ τ for every twosimplices σ, τ ∈ L .First, let us suppose for contradiction that p ( σ ) is affinely dependent forsome σ ∈ L . Then by Radon’s lemma (Lemma 2.1) there are two disjointaffinely independent subsets A, B ⊂ p ( σ ) with intersecting convex hulls. Thenwe have A = p ( α ) and B = p ( β ) for disjoint simplices α, β ∈ L . But we have p ( v ) ∈ C α for all v ∈ α , hence D α = conv( p ( α )) ⊆ C α , and similarly D β ⊆ C β .Then C α ∩ C β ⊇ D α ∩ D β = ∅ , and this contradicts the assumption that the C σ form a representation of N ( L ). So each p ( σ ) is affinely independent.Next, let σ, τ ∈ L . We clearly have D σ ∩ τ ⊆ D σ ∩ D τ . To prove the reverseinclusion, we assume for contradiction that there is some x ∈ ( D σ ∩ D τ ) \ D σ ∩ τ .Lemma 2.2 provides disjoint σ ′ ⊆ σ and τ ′ ⊆ τ with D σ ′ ∩ D τ ′ = ∅ , and this isa contradiction as above. (cid:3) d -Collapsibility of the Nerve Here we prove Proposition 1.3.Let us assume that the ground set of F is [ n ]. Let us fix an arbitrary linearordering ≤ on F . The nerve K = N ( F ) consists of all intersecting subfamiliesof F . For i = 1 , , . . . , n let K i consist of all intersecting families G ∈ K withmin T G = i (so the K i form a partition of K ).Let us consider a G ∈ K i . Each of the elements 1 , , . . . , i − T G by at least one G ∈ G . Let us define the minimal exclusion sequence mes( G ) = ( G , G , . . . , G i − ) as follows. First we choose G as the smallest set of G with 1 G . Having already defined sets G , . . . , G j − ∈ G (not necessarily alldistinct), we define G j as follows: If at least one of the sets among G , . . . , G j − avoids the element j , we let G j be such a G k with the smallest possible k . Inthis case we call G j old at j . On the other hand, if all of G , . . . , G j − contain j , then we let G j be the smallest set of G not containing j , and we call it newat j .Let M ( G ) ∈ K i be the family consisting of all sets G j that occur in mes( G ).In particular, for i = 1 we have M ( G ) = ∅ for all G ∈ K . It is easily seen thatwe always have | M ( G ) | ≤ d (indeed, G covers at most d − , , . . . , i −
1, and only these elements may contribute G j ’s distinct from G ).We also note that mes( M ( G )) = mes( G ).We let M i = { M ( G ) : G ∈ K i } and M = S ni =1 M i . The families in M willbe the d -collapsible simplices we will use for d -collapsing the simplicial complex K = N ( F ). We order them first by decreasing i , i.e., M n comes first, then M n − , etc., and within each M i we order the families lexicographically by theirminimal exclusion sequences. Let (cid:22) denote this linear ordering of M . Thisdefines the sequence of elementary collapses.Clearly, each simplex G ∈ K contains at least one simplex of M , namely, M ( G ). It remains to verify that each M ∈ M is contained in a unique maximalsimplex in the simplicial complex obtained from K by collapsing all N ≺ M .We inductively define K M = ( H ∈ K \ [ N ≺M K N : M ⊆ H ) ;as we will see, this is the set of all simplices removed from the current simplicialcomplex by collapsing M .We need to express K M as the set of all simplices of K \ S N ≺M K N thatcontain M and are contained in a suitable maximal simplex T ( M ). As wewill see, the desired T ( M ) can be described as follows (here M ∈ K i andmes( M ) = ( G , . . . , G i − )): T ( M ) = M∪ n F ∈ F : i ∈ F and F > G j for all j F such that G j is new at j o . That is, T ( M ) consists of M plus those sets of F that contain i and satisfymes( M ∪ { F } ) = mes( M ). Clearly T ( M ) ∈ K and M ⊆ T ( M ). We also have M ( T ( M )) = M . 6e let K ′M = {H ∈ K \ S N ≺M K N : M ⊆ H ⊆ T ( M ) } , and by inductionwe prove that K ′M = K M . This will show that T ( M ) is indeed the uniquemaximal simplex containing M . So we consider some M ∈ M i and we assumethat K ′N = K N for all N ≺ M .By just comparing the definitions we immediately obtain K ′M ⊆ K M . Forthe reverse inclusion let us consider an H ∈ K M , and for contradiction let ussuppose that H contains a set F T ( M ). We will exhibit an N ∈ M with N ⊆ H and
N ≺ M ; this will lead to the desired contradiction, since then weeither have
H ∈ K N or H has been collapsed even earlier.By the definition of T ( M ), there can be two reasons for F T ( M ). First, itmight happen that i F . But then min T H > i , and therefore N := M ( H ) ≺M . This leads to a contradiction as explained above, and hence we may assume i ∈ F .Second, letting mes( M ) = ( G , G , . . . , G i − ), there may be some j < i , j F such that G j is new at j and F < G j (we cannot have F = G j sincewe assumed F
6∈ M ). Let j be the smallest possible with this property. Weconsider the family G = { G , G , . . . , G j − , F, G j +1 , . . . , G i − } , which is in K i .Hence N = M ( G ) is in M i , and we also have N ⊆ H . It now suffices to verifythat
N ≺ M . To this end, we check that the first j terms of the sequencemes( N ) are G , G , . . . , G j − , F , since then mes( N ) is indeed lexicographicallysmaller than mes( M ) = ( G , G , . . . , G j − , G j , . . . ).Let us suppose that mes( N ) agrees with ( G , G , . . . , G j − , F ) in the first k − k th terms agree as well. This is clearif k < j and G k is old at k in mes( M ). If k < j and G k is new at k in mes( M ),then k ∈ F or F > G k , for otherwise, we should have taken k instead of j , andhence the k th terms agree in this case too. Finally, for k = j , G j is new at j inmes( M ) by the assumption, so G j is the smallest set in M not containing j .Then F is even smaller such set, and so it comes to the j th position of mes( N ).This concludes the proof of Proposition 1.3. (cid:3) In this section we prove Theorem 1.1(b).First we recall the notion of join of two simplicial complexes K and L . First,assuming that the vertex sets V ( K ) and V ( L ) are disjoint, the join is the sim-plicial complex K ∗ L := { σ ∪ τ : σ ∈ K , τ ∈ L } on the vertex set V ( K ) ∪ V ( L ).If the vertex sets are not disjoint (as will be the case in our application below),we first take isomorphic copies of K and L with disjoint vertex sets and then weform the join as above.The next lemma shows that the Leray number behaves nicely with respectto joins. Lemma 4.1.
For every two nonempty simplicial complexes K and L we have λ ( K ∗ L ) = λ ( K ) + λ ( L ) . roof. This is a simple consequence of the K¨unneth formula for joins˜ H k ( X ∗ Y ) = M i + j = k − ˜ H i ( X ) ⊗ ˜ H j ( Y ) (3)for any two simplicial complexes X and Y , where ˜ H k ( . ) denotes the k -dimensionalreduced homology group (over Q ). The K¨unneth formula in this form can easilybe derived from [Mun84], Example 4 (p. 349) and Exercise 3 (p. 373).For notational convenience we assume V ( K ) ∩ V ( L ) = ∅ , and let λ ( K ) = k and λ ( L ) = ℓ . Then there exist A ⊆ V ( K ) and B ⊆ V ( L ) such that˜ H k − ( K [ A ]) = 0 = ˜ H ℓ − ( L [ B ]). Since ( K ∗ L )[ A ∪ B ] = K [ A ] ∗ L [ B ], (3) showsthat ˜ H k + ℓ − (( K ∗ L )[ A ∪ B ]) = 0, and thus λ ( K ∗ L ) ≥ k + ℓ .On the other hand, every induced subcomplex of K ∗ L has the form K [ A ] ∗ L [ B ]for some A ⊆ V ( K ) and B ⊆ V ( L ), and if ˜ H i ( K [ A ]) = 0 for all i ≥ k and˜ H j ( L [ B ]) = 0 for all j ≥ ℓ , (3) gives ˜ H s ( K [ A ] ∗ L [ B ]) = 0 for all s ≥ k + ℓ , thusshowing λ ( K ∗ L ) ≤ k + ℓ .We cannot say how γ ( . ) behaves under joins, but we can do so for thefollowing related quantity: γ ( K ) := min { d : K has a d -collapsible face } . Lemma 4.2.
For every two simplicial complexes K , L we have γ ( K ∗ L ) = γ ( K ) + γ ( L ) .Proof. Again we assume V ( K ) ∩ V ( L ) = ∅ . It is easily checked that if σ is a k -collapsible face of K and τ is an ℓ -collapsible face of L , then σ ∪ τ is a ( k + ℓ )-collapsible face of K ∗ L , which shows γ ( K ∗ L ) ≤ γ ( K ) + γ ( L ). On the otherhand, every d -collapsible face of K ∗ L is of the form σ ∪ τ , σ ∈ K , τ ∈ L , andone can check that σ is k -collapsible and τ is ℓ -collapsible for some k, ℓ with k + ℓ = d . This gives the reverse inequality. Proof of Theorem 1.1(b).
We let K be the triangulation of the duncehat in Fig. 3. We have λ ( K ) = 2 < γ ( K ) according to Wegner [Weg75], andactually γ ( K ) = 3 because dim K = 2. Then the join K of d copies of K satisfies λ ( K ) = 2 d and γ ( K ) ≥ γ ( K ) = 3 d by Lemmas 4.1 and 4.2. (cid:3) References [AK92] N. Alon and D. Kleitman. Piercing convex sets and the HadwigerDebrunner ( p, q )-problem.
Adv. Math. , 96(1):103–112, 1992.[AKMM02] N. Alon, G. Kalai, J. Matouˇsek, and R. Meshulam. Transversalnumbers for hypergraphs arising in geometry.
Adv. Appl. Math. ,130:2509–2514, 2002.[Ame96] N. Amenta. A short proof of an interesting Helly-type theorem.
Discrete Comput. Geom. , 15:423–427, 1996.8B´ar82] I. B´ar´any. A generalization of Carath´eodory’s theorem.
DiscreteMath. , 40:141–152, 1982.[DGK63] L. Danzer, B. Gr¨unbaum, and V. Klee. Helly’s theorem and itsrelatives. In
Convexity , volume 7 of
Proc. Symp. Pure Math. , pages101–180. American Mathematical Society, Providence, 1963.[Eck93] J. Eckhoff. Helly, Radon and Carath´eodory type theorems. In P. M.Gruber and J. M. Wills, editors,
Handbook of Convex Geometry .North-Holland, Amsterdam, 1993.[Flo34] A. Flores. ¨Uber n -dimensionale Komplexe die im R n +1 absolutselbstverschlungen sind. Ergeb. Math. Kolloq. , 6:4–7, 1932/1934.[Hat01] A. Hatcher.
Algebraic Topology . Cambridge UniversityPress, Cambridge, 2001. Electronic version available at http://math.cornell.edu/~hatcher .[Hel23] E. Helly. ¨Uber Mengen konvexer K¨orper mit gemeinschaftlichenPunkten.
Jahresbericht Deutsch. Math. Verein. , 32:175–176, 1923.[Hel30] E. Helly. ¨Uber Systeme von abgeschlossenen Mengen mit gemein-schaftlichen Punkten.
Monaths. Math. und Physik , 37:281–302,1930.[Kal84] G. Kalai. Characterization of f-vectors of families of convex sets in R d . I: Necessity of Eckhoff’s conditions. Isr. J. Math. , 48:175–195,1984.[Kal86] G. Kalai. Characterization of f -vectors of families of convex setsin R d . II: Sufficiency of Eckhoff’s conditions. J. Combin. Theory,Ser. A , 41:167–188, 1986.[KL79] M. Katchalski and A. Liu. A problem of geometry in R n . Proc.Amer. Math. Soc. , 75:284–288, 1979.[KM05] G. Kalai and R. Meshulam. A topological colorful Helly theorem.
Adv. Math. , 191(2):305–311, 2005.[KM07] G. Kalai and R. Meshulam. Leray numbers of projections and atopological Helly type theorem. Manuscript, The Hebrew Univer-sity of Jerusalem, 2007.[Lov74] L. Lov´asz. Problem 206.
Matematikai Lapok , 25:181, 1974.[Mat02] J. Matouˇsek.
Lectures on Discrete Geometry . Springer, New York,2002.[Mun84] J. R. Munkres.
Elements of Algebraic Topology . Addison-WesleyPub., New York, 1984.[Rad21] J. Radon. Mengen konvexer K¨orper, die einen gemeinsamen Punktenthalten.
Math. Ann. , 83:113–115, 1921.9vK32] R. E. van Kampen. Komplexe in euklidischen R¨aumen.