Leray Complexes - Combinatorics and Geometry

Roy MeshulamTechnion – Israel Institute of Technology

Joint work with Gil Kalai

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 Caratheodory Theorems

◮ A Topological Matroidal Helly Theorem

Helly’s Theorem (1913)

K be a finite family of convex sets in Rd .K1 ∩ · · · ∩ Kd+1 6= ∅ for all K1, . . . ,Kd+1 ∈ K.Then

⋂

K∈K K 6= ∅.

What is the role of convexity?

Good Covers

X a simplicial complexX1, . . . ,Xn subcomplexes of X

F = {X1, . . . ,Xn} is a good cover if for all σ ⊂ [n] = {1, . . . , n}⋂

i∈σ Xi is either empty or contractible.

Example: If X1, . . . ,Xn are convex, then F is a good cover

Helly’s Topological Theorem (1930)

Let F be a finite good cover in Rd . If F1 ∩ · · · ∩ Fd+1 6= ∅ for allF1, . . . ,Fd+1 ∈ F , then

⋂

F∈F F 6= ∅.

Nerves

F = {F1, . . . ,Fn} family of setsN(F) Nerve of F :Vertices: [n]Simplices: σ ⊂ [n] such that Fσ = ∩i∈σFi 6= ∅

Nerve Lemma (Borsuk)

If F = {X1, . . . ,Xn} is a good cover then N(F) ≃⋃n

i=1 Xi

d -Representable Complexes

X is d-representable X ∈ Kd ifX = N(F) for some family F of convex sets in Rd .

Example: 1-Representable Complexes

K1 = flag complexes of interval graphs

1-representable 1-representablenot

Helly Type Theorems and Nerves

Helly type theorems for families of convex sets K can often beformulated in terms of the nerve N(K), e.g.

Nerve Version of Helly’s Theorem

X ∈ Kd on vertex set V .If {v1, . . . , vd+1} ∈ X for all v1, . . . , vd+1 ∈ V ,then V ∈ X .

Problem: Can the assumption of d-representability be relaxed?The class of d-Leray complexes provides a natural framework forformulating (and sometimes proving) topological Helly typetheorems.

d -Leray Complexes

X is d-Leray X ∈ Ld if Hi (Y ) = 0 forall 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

L1 = flag complexes of chordal graphs

1-representablenot

but 1-Leray

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 = X1 → X2 → · · · → Xm = ∅

Cd= The family of d-collapsible complexes

Representability, Collapsibility and the Leray Condition

Wegner’s Theorem (1975)

Kd ⊂ Cd ⊂ Ld

The righthand side follows from the fact that a d-collapse does noteffect Hi for i ≥ d . The lefthand side Kd ⊂ Cd is a fundamentalresult that underlies many extensions of Helly’s Theorem.For d ≥ 2, the family Ld is larger then Cd .Example: The dunce hat is 2-Leray but not 2-collapsible.

1

1 1

3

3

2

2

3 2

DUNCE HAT

Why do we care about d -Leray complexes ?

◮ Leray number L(X ) is a natural combinatorial-topologicalmeasure of the complexity of X . It can be thought of as the”hereditary homological dimension” of X . As such, it oftenreflects certain deep combinatorial properties of X .E.g. it is known [Alon, Kalai, Matousek, M] that the coveringnumber τ(F) of a hypergraph F is bounded by a function ofits fractional covering number τ∗(F) and the Leray number ofits nerve L(N(F)).

◮ L(X ) comes up in a commutative algebra context as theCastelnuovo-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 othertopological invariants).

Helly Numbers

Helly number h(F) of a family of sets F isthe minimal h such that if a finite subfamily K ⊂ F satisfies⋂

K′ 6= ∅ for all K′ ⊂ K of cardinality ≤ h, then⋂

K 6= ∅.

Examples:

h( the family of all subtrees of a fixed tree ) = 2

h( the family of all convex sets in Rd ) = d + 1

h( the family of all lattice convex sets in Rd ) = 2d

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 Rd then h(F) ≤ d + 1.

Proof: Let F ′ ⊂ F . By Borsuk Nerve Lemma

N(F ′) ≃⋃

F∈F ′

F ⊂ Rd .

Therefore Hi(N(F ′)) = 0 for all i ≥ d . Hence

h(F) ≤ L(N(F)) + 1 ≤ d + 1 .

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 Rd .If F ∈ P(G, r), then h(F) ≤ r(d + 1).

Example for r = d = 2:

F F F F F F1 2 3 4 5 6

Every 5 intersect, but all 6 do not intersect.

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 Rd 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 numberof a projection of a complex.

Notation

V1 ∗ · · · ∗ Vm Join of 0-dimensional complexes∆m−1 (m-1)-dimensional simplex

π : V1 ∗ · · · ∗ Vm → ∆m−1 The natural projection

V1 ∗ · · · ∗ Vm

↓π

∆m−1

Leray Numbers of Projections

X subcomplex of V1 ∗ · · · ∗ Vm

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

Proof of Topological Helly for Unions

G a good cover, F = {F1, . . . ,Fm} ∈ P(G, r)Fi = Gi1 ∪ · · · ∪ Giri , Gij ∩ Gij ′ = ∅ , ri ≤ r

Vi = {Gi1, . . . ,Giri} 0-dimensional complex

X = N({Gij}) ⊂ V1 ∗ · · · ∗ Vm

↓π ↓ππ(X ) = N(F) ⊂ ∆m−1

F ∈ P(G, r)⇓

|π−1(y)| ≤ r

for all y ∈ |N(F)|

V V V1 2 m

1 2 m3

3V

y

X

(X)

The Image Computing Spectral Sequence

π : X → Y a simplicial map, maxy∈|Y | |π−1(y)| < ∞

The Multiple Point Set is:

Dk = {(x1, . . . , xk) ∈ X k : π(x1) = . . . = π(xk)}

The symmetric group Sk acts on Dk , hence on H∗(Dk ; Q).

The Alternating part of H∗(Dk ; Q) is:

Alt H∗(Dk ; Q) = {c ∈ H∗(D

k ; Q) : σc = sign(σ)c , ∀σ ∈ Sk} .

Theorem [Goryunov-Mond]

There exists a spectral sequence {E rp,q} ⇒ H∗(Y ; Q) such that

E 1p,q = AltHq(Dp+1; Q) .

The Homology of Dk

X1, . . . ,Xk subcomplexes of V1 ∗ · · · ∗ Vm .The Generalized Multiple Point Set is:

D(X1, · · · ,Xk) =

{(x1, . . . , xk) ∈ X1 × · · · × Xk : π(x1) = . . . = π(xk)}

Proposition 1 [KM]:

Hj (D(X1, · · · ,Xk)) = 0 for j ≥∑k

i=1 L(Xi ).In particular, Hj (D

k) = 0 for j ≥ kL(X ).

Proposition 2:

Alt H∗(Dk) = 0 for k > r = maxy∈Y |π−1(y)|.

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 thatE 1

p,q = AltHq(Dp+1) = 0 if p + q ≥ rL(X ) + r − 1.

Case 1:p ≥ r ⇒ E 1

p,q = 0 by Proposition 2.

Case 2:p ≤ r − 1 ⇒ q ≥ rL(X ) ≥ (p + 1)L(X )then Hq(D

p+1) = 0 by Proposition 1. Hence E 1p,q = 0.

Application: Leray Numbers of Intersections

X1, . . . ,Xk complexes on V = {1, . . . ,m}Taking V1 = {1}, . . . ,Vm = {m}, it follows that

D(X1, . . . ,Xk) ∼=

k⋂

i=1

Xi

Therefore, Proposition 1 implies:

Corollary [KM]: L(⋂k

i=1 Xi) ≤∑k

i=1 L(Xi )

Example:

X1 = ∂∆k ∗ ∆ℓ L(X1) = k

X2 = ∆k ∗ ∂∆ℓ L(X2) = ℓ

X1 ∩ X2 = ∂∆k ∗ ∂∆ℓ L(X1 ∩ X2) = k + ℓ

Application to Commutative Algebra

Betti NumbersM finitely generated graded module over S = K[x1, . . . , xn].S(−j) = S with shifted grading: S(−j)k = Sk−j .Choose a minimal free resolution

0 → Fr → · · · → F0 → M → 0

with finitely generated graded Fi = ⊕jS(−j)βij .

βij = Betti Numbers of M

Castelnuovo-Mumford Regularity

reg(M) = max{j − i : βij 6= 0}

Simplicial Complexes and Monomial Ideals

X simplicial complex on [n]

1

2

3

4

X

IX = ideal generated by∏

i∈A xi for all A 6∈ X

IX = (x1x4, x2x3x4)

Hochster’s Formula:

βij(IX ) =∑

|W |=j

dim Hj−i−2(X [W ])

Corollary: reg(IX ) = L(X ) + 1

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, thenI = IX , J = IY for some simplicial complexes X ,Y , and thetheorem is equivalent to L(X ∩ Y ) ≤ L(X ) + L(Y ).The general case follows from the squarefree case by polarization.

The Colorful Helly Theorem

Helly’s Theorem

F a finite family of convex sets in Rd .F1 ∩ · · · ∩ Fd+1 6= ∅ for all F1, . . . ,Fd+1 ∈ K.Then

⋂

F∈F F 6= ∅.

Colorful Helly Theorem [Lovasz]:

F1, . . . ,Fd+1 finite families of convex sets in Rd .⋂d+1

i=1 Fi 6= ∅ for all F1 ∈ F1, . . . ,Fd+1 ∈ Fd+1.Then

⋂

F∈FiF 6= ∅ for some 1 ≤ i ≤ d + 1.

F1 · · · Fd+1

The Colorful Caratheodory Theorem

Caratheodory’s Theorem

A a finite set of points in Rd and x ∈ conv(A).Then there exists a subset A′ ⊂ A such that|A′| ≤ d + 1 and x ∈ conv(A′).

Colorful Caratheodory Theorem [Barany]:

A1,A2, . . . Ad+1 finite sets of points in Rd , x ∈ ∩d+1i=1 conv(Ai ).

Then there exist a1 ∈ A1, . . . , ad+1 ∈ Ad+1 such thatx ∈ conv{a1, a2, . . . , ad+1}

Applications of the Colorful Caratheodory Theorem

Tverberg’s Theorem [with Sarkaria’s Proof]

A ⊂ Rd |A| = (k − 1)(d + 1) + 1Then there exists a partition A = A1 ∪ · · · ∪ Ak

such that conv(A1) ∩ · · · ∩ conv(Ak) 6= ∅

Weak ǫ-Nets [Alon, Barany, Furedi, Kleitman]

For any family G of convex sets in Rd

τ(G) ≤ Cd · τ∗(G)d+1

Many Intersecting Simplices [Barany]

Any n points in Rd contain at least cd

(

nd+1

)

intersecting simplices.

Matroidal Helly Theorem

Colorful Helly - Nerve Version

X ∈ Kd on vertex set V = V1 ∪ · · · ∪ Vd+1

{v1, . . . , vd+1} ∈ X for all v1 ∈ V1, . . . , vd+1 ∈ Vd+1

Then Vi ∈ X for some 1 ≤ i ≤ d + 1

V1 · · · Vd+1

Matroidal Helly [KM]:

X ∈ Cd on vertex set V , M matroid on V , M ⊂ X

Then there exists a σ ∈ X such that ρ(V − σ) ≤ d .

Topological Matroidal Helly Theorem

Theorem [KM]:

X ∈ Ld 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 theexistence of colorful simplices

◮ Combinatorial Alexander Duality

Covering and Fractional Covering

F a hypergraph on V

A Cover of F is a subsetS ⊂ V such that S ∩ F 6= ∅ for all F ∈ F .The Covering Number τ(F) isthe minimal cardinality of a cover of F .

A Fractional Cover of F is a functionf : V → R+ such that

∑

v∈F f (v) ≥ 1 for all F ∈ F .The Fractional Covering Number τ∗(F) isminf

∑

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

Covering vs. Fractional Covering Numbers

F a hypergraph on V

deg(v) = |{F ∈ F : v ∈ F}|∆(F) = maxv∈V deg(v) = dimN(F) + 1.

Theorem [Lovasz]: τ(F) ≤ τ∗(F)(1 + log ∆(F)) .

Qualitatively:τ(F) ≤ F1(τ

∗(F), dim N(F))

Theorem [AKMM]: For any d there exist constants c1(d), c2(d)such that if L(N(F)) = d then τ(F) ≤ c1(d)τ∗(F)c2(d) .

Qualitatively:τ(F) ≤ F2(τ

∗(F),L(N(F)))