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 ≈ ℓ B branched branched ≈ ≈ ≡ q � � A B Unravel short cycles and add copies 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 ≈ ℓ A B C unbranched unbranched ≈ C ≈ C ≡ q � � A B Unravel short cycles 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 ] = { ε v } . ◮ ‘Free’ group over E : F E = { reduced words over E } ◮ Short words: F E ,ℓ = { w ∈ F E | | w | ≤ ℓ } ◮ G ℓ -acyclic iff ... iff π [ G , v ] ∩ F E ,ℓ = { ε 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. one of the following equivalent conditions holds: iso ◮ p : star ( � v ) − → 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. one of the following equivalent conditions holds: iso ◮ p : star ( � v ) − → 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 [ � v ]) = { ε v } G , �

Motivation and Proviso Graphs Hypergraphs Final result/Summary Covers of graphs Definition A cover of G is a homomorphism p : � G → G s.t. one of the following equivalent conditions holds: iso ◮ p : star ( � v ) − → 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 [ � v ]) ∩ F E , 5 = { ε v } G , �

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 = � ( g e ) e ∈ E � with g e g e = 1 2. Build the product G × G = ( V × G , E ′ ) G V × { gg e } V × { g } e V v u 3. Use natural hom. ϕ : F E → G to read of cycles: p ∗ ( π [ G × G , ( v , g )]) = π [ G , v ] ∩ ker( ϕ ) ◮ ker( ϕ ) = { ε v } � acyclic cover, ◮ finite G with ker( ϕ ) ∩ F E ,ℓ = { ε v } � 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 ⊆ P 2 ( V ) ◮ H = ( V , S ) , S ⊆ P fin ( V ) ◮ one notion of acyclicity ◮ α α -/ β -/ γ -/Berge- acyclicity α ◮ star G ( v ) always acyclic ◮ star H ( v ) in general cyclic ◮ cycles ◮ ? Results proposal for hypercycles cycles: only for locally acyclic H simple connectivity = acyclicity: finite, ℓ -acyclic covers: only 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 ] π ( | H | , v ) ≃ π [ H , v ] ◮ simple conn. = acyclicity ◮ simple conn. � = acyclicity! Results proposal for hypercycles cycles: only for locally acyclic H simple connectivity = acyclicity: finite, ℓ -acyclic covers: only 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 ◮ finite, ℓ -acyclic covers do covers not exist! Results proposal for hypercycles cycles: only for locally acyclic H simple connectivity = acyclicity: finite, ℓ -acyclic covers: only 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 = ( s i ) i ∈ Z n ⊆ S is a cycle if s i ∩ s i +1 � = ∅ and there is no s ∈ S and j ∈ Z n s.t. s j ∩ s j +1 , s j +1 ∩ s j +2 , s j +2 ∩ s j +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 star H ( 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. iso p : star ( � v ) − → star ( v ) f.a. p ( � v ) = v . unbranched cover d b r a n c h e . . . . . . c o 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 [ � v ]) ⇔ NC G ( R ) ∩ π [ G , v ] ⊆ p ∗ ( π [ � G , � G , � v ]) For � G = G × G with G = F E / N : p ∗ ( π [ � G , ( v , g )]) = ker( ϕ ) ∩ π [ G , v ] = N ∩ π [ G , v ] Take N = NC ( R ): G , ( g , v )]) = NC ( R ) ∩ π [ G , v ] ! p ∗ ( π [ � = NC G ( 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 ℓ -simply connected covers notion of hypercycles acyclicity = simple conn. + local acyclicity ℓ -acyclic covers for locally acyclic H