Acyclicity, Simple Connectivity and Covers of Hypergraphs Julian - - PowerPoint PPT Presentation

Motivation and Proviso Graphs Hypergraphs Final result/Summary

Acyclicity, Simple Connectivity and Covers of Hypergraphs

Julian Bitterlich bitterlich@mathematik.tu-darmstadt.de

Fachbereich Mathematik TU-Darmstadt

May 4, 2017

Motivation and Proviso Graphs Hypergraphs Final result/Summary Characterisation theorems in model theory

Proving Characterisation Theorems via the Upgrading Diagram A B

• A
• B

≈ℓ ≈ ≈ ≡q Unravel short cycles and add copies branched branched From now on: cover means unbranched cover!

Motivation and Proviso Graphs Hypergraphs Final result/Summary Characterisation theorems in model theory

Proving Characterisation Theorems via the Upgrading Diagram A B

• A
• B

≈ℓ

C

≈C ≈C ≡q Unravel short cycles unbranched unbranched From now on: cover means unbranched cover!

Motivation and Proviso Graphs Hypergraphs Final result/Summary Acyclicity of graphs

◮ Walk in graph G = (V , E): sequence of

edges that ‘fit’ together without backtracking

◮ Cycle: walk that starts and ends at the same

node

◮ ‘Fundamental group’: π[G, v] = {cycles at v} ◮ G acyclic iff ... iff π[G, v] = {εv}. ◮ ‘Free’ group over E: FE = {reduced words over E} ◮ Short words: FE,ℓ = {w ∈ FE | |w| ≤ ℓ} ◮ G ℓ-acyclic iff ... iff π[G, v] ∩ FE,ℓ = {εv}

Motivation and Proviso Graphs Hypergraphs Final result/Summary Covers of graphs

Definition

A cover of G is a homomorphism p : G → G s.t.

• ne of the following equivalent conditions holds:

◮ p : star(

v)

iso

− → star(v) f.a. p( v) = v,

◮ walks in G have unique lifts (up to starting point), ◮ the natural projection p∗ : π[

G, v] → π[G, v] is injective. p∗(π[ G, v]) describes which cycles do not get unfolded! p∗(π[ G, v]) = π[G, v]

Motivation and Proviso Graphs Hypergraphs Final result/Summary Covers of graphs

Definition

A cover of G is a homomorphism p : G → G s.t.

• ne of the following equivalent conditions holds:

◮ p : star(

v)

iso

− → star(v) f.a. p( v) = v,

◮ walks in G have unique lifts (up to starting point), ◮ the natural projection p∗ : π[

G, v] → π[G, v] is injective. p∗(π[ G, v]) describes which cycles do not get unfolded! p∗(π[ G, v]) = {εv}

Motivation and Proviso Graphs Hypergraphs Final result/Summary Covers of graphs

Definition

A cover of G is a homomorphism p : G → G s.t.

• ne of the following equivalent conditions holds:

◮ p : star(

v)

iso

− → star(v) f.a. p( v) = v,

◮ walks in G have unique lifts (up to starting point), ◮ the natural projection p∗ : π[

G, v] → π[G, v] is injective. p∗(π[ G, v]) describes which cycles do not get unfolded! p∗(π[ G, v]) ∩ FE,5 = {εv}

Motivation and Proviso Graphs Hypergraphs Final result/Summary Construction of finite, highly acyclic covers

Easy recipe for constructing covers of G = (V , E):

• 1. Take E-group, i.e., G = (ge)e∈E with gege = 1
• 2. Build the product G × G = (V × G, E ′)

V G v u

V × {g} V × {gge} e

• 3. Use natural hom. ϕ: FE → G to read of cycles:

p∗(π[G × G, (v, g)]) = π[G, v] ∩ ker(ϕ)

◮ ker(ϕ) = {εv} acyclic cover, ◮ finite G with ker(ϕ) ∩ FE,ℓ = {εv} finite, ℓ-acyclic cover.

Motivation and Proviso Graphs Hypergraphs Final result/Summary Geometric realisations

Geometric realisation |G| of G: glue points and line segments: Topological space X, x ∈ X: π(X, x) = loops at x/homotopy.

◮ Fundamental group becomes topological fundamental group

π[G, v] ≃ π(|G|, v),

◮ combinatorial cover induces topological cover.

Motivation and Proviso Graphs Hypergraphs Final result/Summary A comparison

Graphs Hypergraphs

◮ G = (V , E), E ⊆ P2(V ) ◮ one notion of acyclicity ◮ starG(v) always acyclic ◮ cycles ◮ H = (V , S), S ⊆ Pfin(V ) ◮ α

α α-/β-/γ-/Berge- acyclicity

◮ starH(v) in general cyclic ◮ ?

Results cycles: simple connectivity = acyclicity: finite, ℓ-acyclic covers: proposal for hypercycles

• nly for locally acyclic H
• nly for locally acyclic H,

in general we get finite, ℓ-simply conn. covers

Motivation and Proviso Graphs Hypergraphs Final result/Summary A comparison

Graphs Hypergraphs π(|G|, v) ≃ π[G, v]

◮ simple conn. = acyclicity

π(|H|, v) ≃ π[H, v]

◮ simple conn. = acyclicity!

Results cycles: simple connectivity = acyclicity: finite, ℓ-acyclic covers: proposal for hypercycles

• nly for locally acyclic H
• nly for locally acyclic H,

in general we get finite, ℓ-simply conn. covers

Motivation and Proviso Graphs Hypergraphs Final result/Summary A comparison

Graphs Hypergraphs

◮ existence of finite, ℓ-acyclic

covers

◮ finite, ℓ-acyclic covers do

not exist! Results cycles: simple connectivity = acyclicity: finite, ℓ-acyclic covers: proposal for hypercycles

• nly for locally acyclic H
• nly for locally acyclic H,

in general we get finite, ℓ-simply conn. covers

Motivation and Proviso Graphs Hypergraphs Final result/Summary A proposal for a ‘hypercycle’

α witness that α is not a hypercycle

Definition

A sequence α = (si)i∈Zn ⊆ S is a cycle if si ∩ si+1 = ∅ and there is no s ∈ S and j ∈ Zn s.t. sj ∩ sj+1, sj+1 ∩ sj+2, sj+2 ∩ sj+3 ⊆ s.

Theorem

H is ℓ-acyclic iff it has no cycles of length less or equal than ℓ.

Motivation and Proviso Graphs Hypergraphs Final result/Summary Simple Connectivity

Geometric realisation |H| of a hypergraph H by gluing points and simplices along their common faces:

Definition

H is locally acyclic if starH(v) is acyclic for every v ∈ V .

Theorem

For connected, locally acyclic H t.f.a.e.

◮ H is simply connected, ◮ H is acyclic.

Motivation and Proviso Graphs Hypergraphs Final result/Summary Covers of Hypergraphs

Definition

A cover of H is a homomorphism p : H → H s.t. p : star( v)

iso

− → star(v) f.a. p( v) = v.

unbranched cover

. . . . . .

b r a n c h e d c

• v

e r finite unbranched cover

Motivation and Proviso Graphs Hypergraphs Final result/Summary Unbranched covers of hypergraphs

Theorem

Every hypergraph has a unique simply connected cover and every finite hypergraph has finite ℓ-simply connected covers.

Proof.

Idea: Take cover p : G → G of the Gaifman graph of H.

• 1. Put all cycles that may not be unraveld in a set R.
• 2. Keep cycles in R intact: R ∩ π[G, v] ⊆ p∗(π[

G, v]).

• 3. Define

H on G. α . . . . . .

Motivation and Proviso Graphs Hypergraphs Final result/Summary How to keep cycles intact

Task: Construct a cover p : G → G s.t. R ∩ π[G, v] ⊆ p∗(π[ G, v]) and p∗(π[ G, v]) is as small as possible Observation: R ∩ π[G, v] ⊆ p∗(π[ G, v]) ⇔ NCG(R) ∩ π[G, v] ⊆ p∗(π[ G, v]) For G = G × G with G = FE/N: p∗(π[ G, (v, g)]) = ker(ϕ) ∩ π[G, v] = N ∩ π[G, v] Take N = NC(R): p∗(π[ G, (g, v)]) = NC(R) ∩ π[G, v] ! = NCG(R) ∩ π[G, v] ‘ℓ-acyclic’ and finite: Analogous with Marshall Hall’s Theorem

Motivation and Proviso Graphs Hypergraphs Final result/Summary R-granular covers

Definition (Core Notion)

Let R be a set cycles in G. A cover ( G, p) is R-granular if cycles in R lift to cycles, i.e., R ∩ π[G, v] ⊆ p∗(π[ G, v]).

Theorem

There is a unique ‘acyclic’ R-granular cover. There are finite, ‘ℓ-acyclic’ R-granular covers, provided G and R are finite. · · · · · ·

Motivation and Proviso Graphs Hypergraphs Final result/Summary Final result/Summary

Graphs ℓ-acyclic granular covers Hypergraphs notion of hypercycles ℓ-simply connected covers acyclicity = simple conn. + local acyclicity ℓ-acyclic covers for locally acyclic H