Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs: basic concepts and recent results Paola Cristofori University of Modena and Reggio Emilia (Italy) Conference “Random Geometry and Physics” Paris, October 17-21, 2016 Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The PL category Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The PL category Objects PL n -manifolds = triangulated (topological) manifolds s.t. each vertex has a neighbourhood whose boundary is PL-isomorphic either to the boundary of an n-simplex or to an ( n − 1) -simplex Morphisms Piecewise Linear (PL) maps = induced by simplicial maps Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group Pseudotriangulations An n -pseudocomplex is a finite collection of n -simplices (together with their faces) such that two n -simplices may have a union of faces in common. A (pseudo)triangulation of a PL n -manifold M is an n -pseudocomplex K whose points form a topological space | K | , PL-isomorphic to M Möbius strip Torus Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group colored triangulations and colored graphs K pseudocomplex triangulating a closed PL n -manifold M n ξ : S 0 ( K ) → ∆ n = { 0 , 1 , . . . , n } ( vertex-labelling ) injective on each n -simplex of K ( K , ξ ) = is a colored triangulation of M n Remark: If M n = | K | , then ( K ′ , ξ ) is a colored triangulation, where K ′ first barycentric subdivision of K ξ ( v ) = r iff v barycenter of an r -simplex of K Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group colored triangulations and colored graphs Γ = ( V (Γ) , E (Γ)) 1-skeleton of the dual cellular complex of K (regular graph of degree n + 1); γ : E (Γ) → ∆ n (edge-coloration) defined by: γ ( e ) = c if e ∈ E (Γ) is dual to an ( n − 1)-simplex of K having no c -labelled vertex. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group The “dual” graph Γ( K ): an example K 1 0 2 1 0 1 0 A colored triangulation of the torus Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group The “dual” graph Γ( K ): an example K 1 0 B A C 2 1 0 F D E 1 0 take a vertex v ( σ ) for each n -simplex σ of K Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group B K Γ 0 1 A C B A C 2 1 0 F D E F D 1 0 E join two vertices v ( σ ) and v ( σ ′ ) with a c -colored edge ( c ∈ ∆ n ) iff σ and σ ′ have in common the ( n − 1)-dimensional face opposite to their c -colored vertices. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group B K Γ 0 1 A C B A C 2 1 0 F D E F D 1 0 E join two vertices v ( σ ) and v ( σ ′ ) with a c -colored edge ( c ∈ ∆ n ) iff σ and σ ′ have in common the ( n − 1)-dimensional face opposite to their c -colored vertices. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group B K Γ 0 1 A C B A C 2 1 0 F D E F D 1 0 E join two vertices v ( σ ) and v ( σ ′ ) with a c -colored edge ( c ∈ ∆ n ) iff σ and σ ′ have in common the ( n − 1)-dimensional face opposite to their c -colored vertices. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group B K Γ 0 1 A C B A C 2 1 0 F D E F D 1 0 E Γ( K ) is an (n+1)-colored graph i.e. adjacent edges have different colors Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group B K Γ 1 0 A C B A C 2 1 0 F D E F D 1 0 E B 0 1 B 2 A C A C D 1 0 D Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group Inverse construction: the pseudocomplex K (Γ) Let (Γ , γ ) be an ( n + 1)-colored graph, 1) take an n -simplex σ ( x ) for every vertex x ∈ V (Γ), and label its vertices by ∆ n ; 2) if x , y ∈ V (Γ) are joined by a c -colored edge, identify the ( n − 1)-faces of σ ( x ) and σ ( y ) opposite to c -labelled vertices, so that equally labelled vertices coincide. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group ⋆ K (Γ) is an n - pseudomanifold ⋆ (Γ , γ ) represents M n = | K (Γ) | ⋆ If M n is a closed manifold, (Γ , γ ) is called a gem = “graph encoded manifold” of M n . Remark: Γ( K (Γ)) = Γ for any Γ, but K (Γ( K )) = K iff the disjoint star of each vertex of K is strongly connected. Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group CONSEQUENCES: M n = | K (Γ) | is orientable iff Γ is bipartite; Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group CONSEQUENCES: M n = | K (Γ) | is orientable iff Γ is bipartite; ∀B ⊂ ∆ n , with # B = h , there is a bijection between ( n − h )-simplices of K (Γ) whose vertices are labelled by ∆ n − {B} and connected components of h -colored graph Γ B = ( V (Γ) , γ − 1 ( B )) ( B -residues of Γ). In particular: c -labelled vertices of K (Γ) are in bijection with connected components of Γ ˆ c = Γ ∆ n −{ c } . Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group CONSEQUENCES: M n = | K (Γ) | is orientable iff Γ is bipartite; ∀B ⊂ ∆ n , with # B = h , there is a bijection between ( n − h )-simplices of K (Γ) whose vertices are labelled by ∆ n − {B} and connected components of h -colored graph Γ B = ( V (Γ) , γ − 1 ( B )) ( B -residues of Γ). In particular: c -labelled vertices of K (Γ) are in bijection with connected components of Γ ˆ c = Γ ∆ n −{ c } . | K (Γ) | is a (closed) n -manifold if and only if, for every c ∈ ∆ n , each c represents S n − 1 . connected component of Γ ˆ Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory Representing PL manifolds by edge-colored graphs PL invariants via colored graph Moves Crystallizations of 4-manifolds Computation of the fundamental group An ( n + 1)-colored graph (Γ , γ ) is called contracted if ∀ c ∈ ∆ n , either Γ ˆ c is connected or no connected components of Γ ˆ c represents the ( n − 1)-sphere. A crystallization of a closed n -manifold M n is a contracted gem (Γ , γ ) of M n . By duality this is equivalent to requiring K (Γ) to have exactly n + 1 vertices ( = minimum possible number of vertices). Paola Cristofori Representing PL manifolds by edge-colored graphs
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