Leray Complexes - Combinatorics and Geometry

Transcription

1 Leray Complexes - Combinatorics and Geometry Roy Meshulam Technion Israel Institute of Technology Joint work with Gil Kalai

2 Plan Introduction Helly s Theorem d-representable, d-collapsible & d-leray complexes Topological Helly Theorem for Unions Leray Numbers of Projections Application to Commutative Algebra Topological Colorful Helly Theorem The Colorful Helly and Carathéodory Theorems A Topological Matroidal Helly Theorem

3 Helly s Theorem (1913) K be a finite family of convex sets in R d. K 1 K d+1 for all K 1,...,K d+1 K. Then K K K. What is the role of convexity?

4 Good Covers X a simplicial complex X 1,...,X n subcomplexes of X F = {X 1,...,X n } is a good cover if for all σ [n] = {1,...,n} i σ X i is either empty or contractible. Example: If X 1,...,X n are convex, then F is a good cover Helly s Topological Theorem (1930) Let F be a finite good cover in R d. If F 1 F d+1 for all F 1,...,F d+1 F, then F F F.

5 Nerves F = {F 1,...,F n } family of sets N(F) Nerve of F: Vertices: [n] Simplices: σ [n] such that F σ = i σ F i Nerve Lemma (Borsuk) If F = {X 1,...,X n } is a good cover then N(F) n i=1 X i

6 d-representable Complexes X is d-representable X K d if X = N(F) for some family F of convex sets in R d. Example: 1-Representable Complexes K 1 = flag complexes of interval graphs 1-representable not 1-representable

7 Helly Type Theorems and Nerves Helly type theorems for families of convex sets K can often be formulated in terms of the nerve N(K), e.g. Nerve Version of Helly s Theorem X K d on vertex set V. If {v 1,...,v d+1 } X for all v 1,...,v d+1 V, then V X. Problem: Can the assumption of d-representability be relaxed? The class of d-leray complexes provides a natural framework for formulating (and sometimes proving) topological Helly type theorems.

8 d-leray Complexes X is d-leray X L d if H i (Y ) = 0 for all induced subcomplexes Y of X and all i d. L(X) =Leray Number of X = min{d : X d Leray} L(X)=0 L(X)=2 Example: 1-Leray Complexes L 1 = flag complexes of chordal graphs not 1-representable but 1-Leray

9 d-collapsible Complexes X simplicial complex, σ X, σ d Suppose σ is contained in a unique maximal face τ X. An Elementary d-collapse: X X {η : σ η τ} 2-coll. 2-coll. A d-collapse is a sequence of elementary d-collapses: X = X 1 X 2 X m = C d = The family of d-collapsible complexes

10 Representability, Collapsibility and the Leray Condition Wegner s Theorem (1975) K d C d L d The righthand side follows from the fact that a d-collapse does not effect H i for i d. The lefthand side K d C d is a fundamental result that underlies many extensions of Helly s Theorem. For d 2, the family L d is larger then C d. Example: The dunce hat is 2-Leray but not 2-collapsible. 1 DUNCE HAT

11 Why do we care about d-leray complexes? Leray number L(X) is a natural combinatorial-topological measure of the complexity of X. It can be thought of as the hereditary homological dimension of X. As such, it often reflects certain deep combinatorial properties of X. E.g. it is known [Alon, Kalai, Matoušek, M] that the covering number τ(f) of a hypergraph F is bounded by a function of its fractional covering number τ (F) and the Leray number of its nerve L(N(F)). L(X) comes up in a commutative algebra context as the Castelnuovo-Mumford regularity of the ring of X. d-collapsibility is a combinatorial reason for being d-leray. However, a complex can be d-leray for other, less tangible, reasons. (This phenomenon is well-known for other topological invariants).

12 Helly Numbers Helly number h(f) of a family of sets F is the minimal h such that if a finite subfamily K F satisfies K for all K K of cardinality h, then K. Examples: h( the family of all subtrees of a fixed tree ) = 2 h( the family of all convex sets in R d ) = d + 1 h( the family of all lattice convex sets in R d ) = 2 d

13 Helly Numbers via Leray Numbers Claim: h(f) L(N(F)) + 1. Example: Helly s Topological Theorem If F is a finite good cover in R d then h(f) d + 1. Proof: Let F F. By Borsuk Nerve Lemma N(F ) F F F R d. Therefore H i (N(F )) = 0 for all i d. Hence h(f) L(N(F)) + 1 d + 1.

14 Helly Theorem for Unions G, F families of sets. F satisfies P(G,r) if for any F F, the intersection F is a union of at most r disjoint sets in G. Theorem [Amenta]: Let G = compact convex sets in R d. If F P(G,r), then h(f) r(d + 1). Example for r = d = 2: F F F F F F Every 5 intersect, but all 6 do not intersect.

15 Topological Helly Theorem for Unions Theorem [KM]: Let G be good cover in some topological space. Then F P(G,r) implies L(N(F)) rl(n(g)) + r 1 Corollary [KM]: If G is a good cover in R d then h(f) L(N(F)) + 1 rl(n(g)) + r r(d + 1) The main ingredient in the proof is a bound on the Leray number of a projection of a complex.

16 Notation V 1 V m m 1 π : V 1 V m m 1 Join of 0-dimensional complexes (m-1)-dimensional simplex The natural projection V 1 V m π m 1

17 Leray Numbers of Projections X subcomplex of V 1 V m Y = π(x) subcomplex of m 1 r = max{ π 1 (y) : y Y } Theorem [KM]: L(Y ) rl(x) + r 1 * * * * * * L(X) = 2 Y = π(x) = 8 L(Y ) = 8 = 3L(X) + 2

18 Proof of Topological Helly for Unions G a good cover, F = {F 1,...,F m } P(G,r) F i = G i1 G iri, G ij G ij =, r i r V i = {G i1,...,g iri } 0-dimensional complex X = N({G ij }) π π(x) = N(F) V 1 V m π m 1 F P(G,r) π 1 (y) r for all y N(F) V 1 V 2 V 3 Vm X y m (X)

19 The Image Computing Spectral Sequence π : X Y a simplicial map, max y Y π 1 (y) < The Multiple Point Set is: D k = {(x 1,...,x k ) X k : π(x 1 ) =... = π(x k )} The symmetric group S k acts on D k, hence on H (D k ; Q). The Alternating part of H (D k ; Q) is: Alt H (D k ; Q) = {c H (D k ; Q) : σc = sign(σ)c, σ S k }. Theorem [Goryunov-Mond] There exists a spectral sequence {E r p,q} H (Y ; Q) such that E 1 p,q = AltH q (D p+1 ; Q).

20 The Homology of D k X 1,...,X k subcomplexes of V 1 V m. The Generalized Multiple Point Set is: D(X 1,,X k ) = {(x 1,...,x k ) X 1 X k : π(x 1 ) =... = π(x k )} Proposition 1 [KM]: H j (D(X 1,,X k )) = 0 for j k i=1 L(X i). In particular, H j (D k ) = 0 for j kl(x). Proposition 2: Alt H (D k ) = 0 for k > r = max y Y π 1 (y).

21 Leray Numbers of Projections - Proof Theorem [KM]: Y = π(x), r = max{ π 1 (y) : y Y }. Then: L(Y ) rl(x) + r 1. Proof: By the Goryunov-Mond sequence, it suffices to show that E 1 p,q = AltH q (D p+1 ) = 0 if p + q rl(x) + r 1. Case 1: p r Ep,q 1 = 0 by Proposition 2. Case 2: p r 1 q rl(x) (p + 1)L(X) then H q (D p+1 ) = 0 by Proposition 1. Hence E 1 p,q = 0.

22 Application: Leray Numbers of Intersections X 1,...,X k complexes on V = {1,...,m} Taking V 1 = {1},...,V m = {m}, it follows that D(X 1,...,X k ) = Therefore, Proposition 1 implies: k i=1 X i Corollary [KM]: L( k i=1 X i) k i=1 L(X i) Example: X 1 = k l X 2 = k l X 1 X 2 = k l L(X 1 ) = k L(X 2 ) = l L(X 1 X 2 ) = k + l

23 Application to Commutative Algebra Betti Numbers M finitely generated graded module over S = K[x 1,...,x n ]. S( j) = S with shifted grading: S( j) k = S k j. Choose a minimal free resolution 0 F r F 0 M 0 with finitely generated graded F i = j S( j) β ij. β ij = Betti Numbers of M Castelnuovo-Mumford Regularity reg(m) = max{j i : β ij 0}

24 Simplicial Complexes and Monomial Ideals X simplicial complex on [n] 2 I X = ideal generated by i A x i for all A X 1 4 I X = (x 1 x 4,x 2 x 3 x 4 ) 3 X Hochster s Formula: β ij (I X ) = dim H j i 2 (X[W ]) W =j Corollary: reg(i X ) = L(X) + 1

25 Regularity of a Sum of Monomial Ideals The following result was conjectured by Terai. Theorem [KM]: If I,J are generated by monomials then reg(i + J) reg(i) + reg(j) 1. Proof: If I and J are generated by squarefree monomials, then I = I X, J = I Y for some simplicial complexes X,Y, and the theorem is equivalent to L(X Y ) L(X) + L(Y ). The general case follows from the squarefree case by polarization.

26 The Colorful Helly Theorem Helly s Theorem F a finite family of convex sets in R d. F 1 F d+1 for all F 1,...,F d+1 K. Then F F F. Colorful Helly Theorem [Lovász]: F 1,..., F d+1 finite families of convex sets in R d. d+1 i=1 F i for all F 1 F 1,...,F d+1 F d+1. Then F F i F for some 1 i d + 1. F 1 F d+1

27 The Colorful Carathéodory Theorem Carathéodory s Theorem A a finite set of points in R d and x conv(a). Then there exists a subset A A such that A d + 1 and x conv(a ). Colorful Carathéodory Theorem [Bárány]: A 1,A 2,...A d+1 finite sets of points in R d, x d+1 i=1 conv(a i). Then there exist a 1 A 1,...,a d+1 A d+1 such that x conv{a 1,a 2,...,a d+1 }

28 Applications of the Colorful Carathéodory Theorem Tverberg s Theorem [with Sarkaria s Proof] A R d A = (k 1)(d + 1) + 1 Then there exists a partition A = A 1 A k such that conv(a 1 ) conv(a k ) Weak ǫ-nets [Alon, Bárány, Füredi, Kleitman] For any family G of convex sets in R d τ(g) C d τ (G) d+1 Many Intersecting Simplices [Bárány] Any n points in R d contain at least c d ( n d+1) intersecting simplices.

29 Matroidal Helly Theorem Colorful Helly - Nerve Version X K d on vertex set V = V 1 V d+1 {v 1,...,v d+1 } X for all v 1 V 1,...,v d+1 V d+1 Then V i X for some 1 i d + 1 V 1 V d+1 Matroidal Helly [KM]: X C d on vertex set V, M matroid on V, M X Then there exists a σ X such that ρ(v σ) d.

30 Topological Matroidal Helly Theorem Theorem [KM]: X L d on vertex set V, M matroid on V, M X Then there exists a σ X such that ρ(v σ) d. Main Ingredients in Proof: Homological Hall Lemma on the existence of colorful simplices Combinatorial Alexander Duality

31 Covering and Fractional Covering F a hypergraph on V A Cover of F is a subset S V such that S F for all F F. The Covering Number τ(f) is the minimal cardinality of a cover of F. A Fractional Cover of F is a function f : V R + such that v F f (v) 1 for all F F. The Fractional Covering Number τ (F) is min f v V f (v) over all fractional covers f. Example: The Complete k-uniform Hypergraph V = [n], F = ( [n]) k τ (F) = n k < τ(f) = n k + 1

32 Covering vs. Fractional Covering Numbers F a hypergraph on V deg(v) = {F F : v F } (F) = max v V deg(v) = dimn(f) + 1. Theorem [Lovász]: τ(f) τ (F)(1 + log (F)). Qualitatively: τ(f) F 1 (τ (F),dim N(F)) Theorem [AKMM]: For any d there exist constants c 1 (d),c 2 (d) such that if L(N(F)) = d then τ(f) c 1 (d)τ (F) c 2(d). Qualitatively: τ(f) F 2 (τ (F),L(N(F)))