Graph-theoretic methods in combinatorial (algebraic) topology Micha - - PowerPoint PPT Presentation

Graph-theoretic methods in combinatorial (algebraic) topology

Micha l Adamaszek

Universit¨ at Bremen Joint work with Jan Hladk´ y and Juraj Stacho

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Combinatorial (algebraic) topology

complexes arising from combinatorial objects, applications of topology to combinatorics, computational aspects, triangulations, face numbers, embeddability, applied algebraic topology, probabilistic topology.

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Flag complexes

If G is a graph, then the clique complex Cl(G) is the simplicial complex whose faces are the cliques (complete subgraphs) of G.

Source: Wikipedia

a.k.a. flag complexes, Vietoris-Rips complexes,

• rder complexes ∆P of posets,

simplicial curvature a la Gromov.

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Complexity of H∗(K)

Problem (Kaibel, Pfetsch, Algorithmic Problems in Polytope Theory) Given a simplicial complex K, presented by the list of maximal faces, what is the complexity of calculating H∗(K) ? K ⇓ · · · ← Cn−1(K) ← Cn(K) ← Cn+1(K) ← · · · ⇓ Hn(K) The first stage seems to require exponential time.

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NP-hardness

NP-hard problems = decision (Yes/No) problems which prov- ably require more than polynomial time (unless P = NP) Take an instance I of your favorite problem P which you already know is NP-hard. Construct a simplicial complex K = K(I) and n ∈ N such that Hn(K) = 0 ⇐ ⇒ I is a Yes-instance The homology problem is then “at least as hard” as P.

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Hyperoctahedral spheres

O0 O1 O2 O3 ΣO2 These are the clique complexes of the graphs: K2 K2,2 K2,2,2 K2,2,2,2 · · ·

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Hyperoctahedral classes in homology of flag complexes

On ֒ → K ⇒ α ∈ Hn(K) some face of On is a maximal face of K ⇒ α = 0 K2, 2, . . . , 2

• n+1

֒ → G ⇒ α ∈ Hn(Cl(G)) some clique in K2, 2, . . . , 2

• n+1

is a maximal clique of G ⇒ α = 0 “n-gadget in G” n = 1

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The main result

Theorem (MA+JS) There is a class of graphs (cochordal), such that For every graph G in the class every group Hn(Cl(G)) is generated by n-gadgets. Given a graph G in the class and an integer n it is NP-hard to decide if G contains an n-gadget.

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Consequences for complexity of H∗(−)

Theorem The following problems are NP-hard Given a graph G and an integer n, decide if Hn(Cl(G)) = 0 (remains NP-hard even restricted to cochordal graphs). Given any simplicial complex K, presented as the list of maximal faces, and an integer n, decide if Hn(K) = 0. Let K = Alexander dual of Cl(G). Hn(K) = H|G|−n−3(Cl(G)), max-faces of K are the complements of non-edges of G.

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Flag spheres

Problem (Hopf) The Euler characteristic of a 2n-dimensional manifold M of non-positive sectional curvature satisfies (−1)nχ(M) ≥ 0. Charney and Davis (1995) develop a local, combinatorial analogue.

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Charney-Davis conjecture

Problem (Charney, Davis) If K is a (2s − 1)-dimensional flag sphere then

• i

(−1 2)ifi(K) ≥ 0. The C.-D. conjecture implies the Hopf conjecture for manifolds with a cubical cell decomposition. Equality holds for the hyperoctahedral spheres, their various subdivisions and... Theorem (Davis, Okun (2001)) If K is a flag 3-sphere with f0 vertices and f1 edges then f1 ≥ 5f0 − 16.

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Face numbers of flag spheres

Combinatorial characterization of f -vectors of flag d-spheres. d = 1

• bvious

d = 2

• bvious

d = 3 known up to possibly a finite number

• f cases, [Davis-Okun, Gal, MA+JH]

d = 4 known, [Davis-Okun, Gal, Nevo- Murai] d = 2s − 1 ≥ 5

• ne non-trivial restriction

f1 ≤ s − 1 2s f 2

0 + f0

for sufficiently large f0 [MA]

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Upper bound for f1 in flag 3-spheres

Thm: If G – graph with n vertices, m edges and K = Cl(G) = S3 then m ≤ 1

4n2 + n.

lkKv = S2, lkKv does not contain the subgraph K3,3, G does not contain the subgraph K1,3,3, (Erd¨

• s) for large n, the maximizer of |E(G)| among K1,3,3-free

graphs is fig for this graph m = 1

4n2 + n and Cl(G) = S1 ∗ S1 = S3.

In the general case use van Kampen – Flores: Cl(K3, . . . , 3

n+1

) ֒ → S2n.

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Background

Theorem (Mantel,Turan) If G is a graph with n vertices, m edges and no triangles then m ≤ 1 4n2 and the maximizer is Kn/2,n/2. K = Cl(G) = S3, f0 = n, f1 = m, f2 = 2(m − n) ≈ n2 Theorem (Stability,Erd¨

• s,Simonovitz,Lovasz)

If G is a graph with n vertices, m ≥ 1

4n2 edges and only ≈ n2

triangles then G is “very similar” to Kn/2,n/2.

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The stability method — a general approach

Suppose G is very dense (m ≥ 1

4n2) and Cl(G) = S3

Stability ⇒ G is similar to Kn/2,n/2 G has extra geometric properties which can be used to show that in fact G must look like fig

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Results

Theorem (MA+JH, conjectured by Gal) If K is a flag 3-sphere with the number of edges close to maximum, precisely 1 4f 2

0 + 1

2f0 + 17 ≤ f1 ≤ 1 4f 2

0 + f0

and f0 is sufficiently large, then K is still a join of two cycles. Theorem (MA) The inequality f1 ≤ s − 1 2s f 2

0 + f0

holds for a large class of (2s − 1)-dimensional weak pseudomanifolds with sufficiently many vertices, including in particular (2s − 1)-dimensional spheres, homology spheres, closed manifolds, homology manifolds and more.

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