Homology of generalized generalized graph homology generalizing to - PowerPoint PPT Presentation

homology of configuration spaces Cooper the chromatic polynomial Homology of generalized generalized graph homology generalizing to configuration spaces simplicial complexes categorification Andrew Cooper joint work with R. Sazdanovi´ c (NCSU) and V. de Silva (Pomona) TOPOSYM 2016 ˇ CVUT July 26, 2016

homology of configuration spaces Cooper the chromatic polynomial graph homology generalizing to simplicial complexes categorification the chromatic polynomial

homology of the chromatic polynomial configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition simplicial complexes categorification The chromatic function P G ( λ ) of the graph G = ( V , E ) is the number of ways, given λ colors, to color each vertex v ∈ V so that adjacent vertices have distinct colors.

homology of the chromatic polynomial configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition simplicial complexes categorification The chromatic function P G ( λ ) of the graph G = ( V , E ) is the number of ways, given λ colors, to color each vertex v ∈ V so that adjacent vertices have distinct colors. Originally introduced by Birkhoff 1912 to prove the Four Color Theorem. (Birkhoff-Lewis proved that 5 colors suffice.)

homology of the chromatic polynomial configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition simplicial complexes categorification The chromatic function P G ( λ ) of the graph G = ( V , E ) is the number of ways, given λ colors, to color each vertex v ∈ V so that adjacent vertices have distinct colors. Originally introduced by Birkhoff 1912 to prove the Four Color Theorem. (Birkhoff-Lewis proved that 5 colors suffice.) Elementary fact P G ( λ ) is polynomial in λ , of degree | V | .

homology of the chromatic polynomial configuration spaces Cooper the chromatic polynomial graph homology generalizing to simplicial complexes Theorem (deletion-contraction formula) categorification Given a simple graph G = ( V , E ) and e ∈ E, let G − e = ( V , E \ { e } ) and G / e be the graph given by contracting e to a point. Then P G = P G − e − P G / e

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification P G does not detect

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness P G does not detect

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) P G does not detect

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) ◮ number of connected components P G does not detect

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) ◮ number of connected components ◮ · · · P G does not detect

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) ◮ number of connected components ◮ · · · P G does not detect ◮ isomorphism type

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) ◮ number of connected components ◮ · · · P G does not detect ◮ isomorphism type ◮ homotopy type

homology of the chromatic polynomial configuration spaces Cooper P G detects combinatorial and topological features the chromatic polynomial ◮ | V | graph homology generalizing to ◮ | E | simplicial complexes categorification ◮ bipartiteness ◮ treeness (if | V | is even) ◮ number of connected components ◮ · · · P G does not detect ◮ isomorphism type ◮ homotopy type A more sophisticated invariant is called for. . . numerical invariants, characteristic polynomial, Tutte polynomial, . . .

homology of configuration spaces Cooper the chromatic polynomial graph homology generalizing to simplicial complexes categorification graph configuration spaces and graph homology

homology of configuration spaces configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition (Fadell, Neuwrith) simplicial complexes categorification Configuration space on a topological space X is the space of n-tuples of distinct points on X � ( x 1 , x 2 , . . . , x n ) ∈ X n � � � x i � = x j if i � = j � Conf ( X , n ) =

homology of configuration spaces configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition (Fadell, Neuwrith) simplicial complexes categorification Configuration space on a topological space X is the space of n-tuples of distinct points on X � ( x 1 , x 2 , . . . , x n ) ∈ X n � � � x i � = x j if i � = j � Conf ( X , n ) = ◮ applications to harmonic analysis, complex analysis, geometry, physics

homology of configuration spaces configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition (Fadell, Neuwrith) simplicial complexes categorification Configuration space on a topological space X is the space of n-tuples of distinct points on X � ( x 1 , x 2 , . . . , x n ) ∈ X n � � � x i � = x j if i � = j � Conf ( X , n ) = ◮ applications to harmonic analysis, complex analysis, geometry, physics ◮ X manifold ⇒ Conf ( X , n ) manifold

homology of configuration spaces configuration spaces Cooper the chromatic polynomial graph homology generalizing to Definition (Fadell, Neuwrith) simplicial complexes categorification Configuration space on a topological space X is the space of n-tuples of distinct points on X � ( x 1 , x 2 , . . . , x n ) ∈ X n � � � x i � = x j if i � = j � Conf ( X , n ) = ◮ applications to harmonic analysis, complex analysis, geometry, physics ◮ X manifold ⇒ Conf ( X , n ) manifold ◮ natural action of S n on Conf ( X , n )

homology of graph configuration space configuration spaces Cooper the chromatic polynomial Conf ( X , n ) graph homology Start with X n , remove all diagonals ∆ ij = { x i = x j } for i � = j . generalizing to simplicial complexes categorification

homology of graph configuration space configuration spaces Cooper the chromatic polynomial Conf ( X , n ) graph homology Start with X n , remove all diagonals ∆ ij = { x i = x j } for i � = j . generalizing to simplicial complexes categorification graph configuration space (Eastwood-Huggett) ◮ G = ( V , E ) graph, X topological space, n = | V | ◮ For each e = [ v i , v j ] ∈ E , set � � � x i = x j � ( x 1 , . . . , x n ) � ⊂ X n ∆ e := � ◮ M G ( X ) := X n \ ∆ e e ∈ E ◮ M K n ( X ) = Conf ( X , n ) ; M · · · · · ( X ) = X n ���� n

homology of graph configuration space configuration spaces Cooper the chromatic polynomial graph homology Theorem (Eastwood-Huggett) generalizing to simplicial complexes The Euler characteristic χ ( M G ( X )) satisfies: categorification χ ( M G ( X )) = χ ( M G − e ( X )) − χ ( M G / e ( X )) Corollary χ ( M G ( X )) = P G ( χ ( X )) .

homology of graph configuration space configuration spaces Cooper the chromatic polynomial graph homology Theorem (Eastwood-Huggett) generalizing to simplicial complexes The Euler characteristic χ ( M G ( X )) satisfies: categorification χ ( M G ( X )) = χ ( M G − e ( X )) − χ ( M G / e ( X )) Corollary χ ( M G ( X )) = P G ( χ ( X )) . The homology H ∗ ( M G ( X )) is a categorification of the value P G ( χ ( X )) .

homology of categorification (with apologies to Plato) configuration spaces Cooper the chromatic polynomial graph homology generalizing to simplicial complexes categorification

homology of categorification (with apologies to Plato) configuration spaces Cooper the chromatic polynomial graph homology generalizing to simplicial complexes categorification

homology of categorification configuration spaces Cooper the (fuzzy, vague) idea the chromatic polynomial Given a structure S , assign a category C ( S ) , a categorification graph homology H : S → Obj ( C ( S )) , a characteristic χ : Obj ( C ( S )) → S so that generalizing to simplicial complexes χ ( H ( s )) = s categorification