Combinatorial algebraic topology of toric arrangements. Emanuele Delucchi (SNSF / Universit´ e de Fribourg) Universit` a di Pisa February 3., 2016
Combinatorial algebraic topology Algebraic Combinatorics topology
Combinatorial algebraic topology Algebraic Combinatorics topology This talk 1. Problem, context Outline: 2. Our tools 3. Our solution
The problem Toric arrangements A toric arrangement in the complex torus T := ( C ∗ ) d is a set A := { K 1 , . . . , K n } of ‘hypertori’ K i = χ − 1 i ( b i ) with χ i ∈ Hom � =0 ( T, C ∗ ) and b i ∈ C ∗ / = 1 / ∈ S 1
The problem Toric arrangements A toric arrangement in the complex torus T := ( C ∗ ) d is a set A := { K 1 , . . . , K n } of ‘hypertori’ K i = { z ∈ T | z a i = b i } with a i ∈ Z d \ 0 and b i ∈ C ∗ For simplicity assume that the matrix [ a 1 , . . . , a n ] has rank d . The complement of A is M ( A ) := T \ ∪ A ,
The problem Toric arrangements A toric arrangement in the complex torus T := ( C ∗ ) d is a set A := { K 1 , . . . , K n } of ‘hypertori’ K i = { z ∈ T | z a i = b i } with a i ∈ Z d \ 0 and b i ∈ C ∗ The complement of A is M ( A ) := T \ ∪ A , Problem: Determine the ring H ∗ ( M ( A ) , Z ).
Context General problem Let X be a complex manifold, A := { L i } i a family of submanifolds of X . Determine the topology of � M ( A ) := X \ L i . i Examples: normal crossing divisors (Deligne), arrangements of hypersur- faces (Dupont), configuration spaces (e.g., Totaro), affine subspace arrange- ments (e.g., Goresky-MacPherson, De Concini-Procesi), toric arrangements, arrangements of hyperplanes, etc.
Context General problem Let X be a complex manifold, A := { L i } i a family of submanifolds of X . Determine the topology of � M ( A ) := X \ L i . i What can combinatorial models tell us? “Exhibit A”: Arrangements of (real) pseudospheres ↔ Oriented matroids.
Context Hyperplanes: Brieskorn A := { H 1 , . . . , H d } : set of (affine) hyperplanes in C d , L ( A ) := {∩ B | B ⊆ A } : (po)set of intersections (reverse inclusion). A L ( A ) X
Context Hyperplanes: Brieskorn A := { H 1 , . . . , H d } : set of (affine) hyperplanes in C d , L ( A ) := {∩ B | B ⊆ A } : (po)set of intersections (reverse inclusion). For X ∈ L ( A ): A X = { H i ∈ A | X ⊆ H i } . A L ( A ) A X X
Context Hyperplanes: Brieskorn A := { H 1 , . . . , H d } : set of (affine) hyperplanes in C d , L ( A ) := {∩ B | B ⊆ A } : (po)set of intersections (reverse inclusion). For X ∈ L ( A ): A X = { H i ∈ A | X ⊆ H i } . A L ( A ) A X X Theorem (Brieskorn 1972). The inclusions M ( A ) ֒ → M ( A X ) induce, for every k , an isomorphism of free abelian groups � ∼ = H k ( M ( A X ) , Z ) → H k ( M ( A ) , Z ) b : − X ∈L ( A ) codim X = k
Context Hyperplanes: The Orlik-Solomon algebra [Arnol’d ‘69, Orlik-Solomon ‘80] H ∗ ( M ( A ) , Z ) ≃ E/ J ( A ), where E : exterior Z -algebra with degree-1 generators e 1 , . . . , e n (one for each H i ); J ( A ): the ideal � � k l =1 ( − 1) l e j 1 · · · � e j l · · · e j k | codim( ∩ i =1 ...k H j i ) = k − 1 �
Context Hyperplanes: The Orlik-Solomon algebra [Arnol’d ‘69, Orlik-Solomon ‘80] H ∗ ( M ( A ) , Z ) ≃ E/ J ( A ), where E : exterior Z -algebra with degree-1 generators e 1 , . . . , e n (one for each H i ); J ( A ): the ideal � � k l =1 ( − 1) l e j 1 · · · � e j l · · · e j k | codim( ∩ i =1 ...k H j i ) = k − 1 � L ( A ) This is fully determined by L ( A ). codim X For instance: � µ L ( A ) (ˆ ( − t ) rk X P ( M ( A ) , t ) = 0 , X ) � �� � X ∈L ( A ) M¨ obius Poin( M ( A ) , t ) = function of L ( A ) 1 + 4 t + 5 t 2 + 2 t 3
Context Toric arrangements Here the role of the intersection poset is played by C ( A ), the poset of layers (i.e. connected components of intersections of the K i ). A : C ( A ):
Context Toric arrangements Here the role of the intersection poset is played by C ( A ), the poset of layers (i.e. connected components of intersections of the K i ). A : C ( A ): Theorem [Looijenga ‘95, De Concini-Procesi ‘05] � ( − t ) rk Y (1 + t ) d − rk Y . µ C ( A ) (ˆ Poin( M ( A ) , Z ) = 0 , Y ) � �� � Y ∈C ( A ) M¨ obius function of C ( A )
Context Toric arrangements [De Concini – Procesi ’05] compute the cup product in H ∗ ( M ( A ) , C ) when the matrix [ a 1 , . . . , a n ] is totally unimodular. [Moci – Settepanella, ’11] Combinatorial models for “thick” arrangements. [Bibby ’14] Q -cohomology algebra of unimodular abelian arrangements [Dupont ’14, ’15] Algebraic model for C -cohomology algebra of complements of hypersurface arrangements in manifolds with hyperplane-like crossings; formality (coming up!), We strive for a (combinatorial) presentation of the integer cohomology ring.
Tools Posets and categories P - a partially ordered set ∆( P ) - the order complex of P (abstract simplicial complex of totally ordered subsets) || P || := | ∆( P ) | its geometric realization a a b c ab ac ( ∅ ) b c || P || P ∆ P
Tools Posets and categories P - a partially ordered set C - a s.c.w.o.l. / “acyclic category” (all invertibles are endomorphisms, all endomorphisms are identities) ∆( P ) - the order complex of P ∆ C - the nerve (abstract simplicial complex (simplicial set of composable chains) of totally ordered subsets) || P || := | ∆( P ) | ||C|| := | ∆ C| its geometric realization its geometric realization a a b c ( ∅ ) ab ac b c || P || C P ∆ P
Tools Posets and categories P - a partially ordered set C - a s.c.w.o.l. / “acyclic category” (all invertibles are endomorphisms, all endomorphisms are identities) ∆( P ) - the order complex of P ∆ C - the nerve (abstract simplicial complex (simplicial set of composable chains) of totally ordered subsets) || P || := | ∆( P ) | ||C|| := | ∆ C| its geometric realization its geometric realization • Posets are special cases of s.c.w.o.l.s; • Every functor F : C → D induces a continuous map || F || : ||C|| → ||D|| . • Quillen-type theorems relate properties of || F || and F .
Tools Face categories Let X be a polyhedral complex. The face category of X is F ( X ), with • Ob( F ( X )) = { X α , polyhedra of X } . • Mor F ( X ) ( X α , X β ) = { face maps X α → X β } Theorem. There is a homeomorphism ||F ( X ) || ∼ = X . [Kozlov / Tamaki] i ( b i ) } with b i ∈ S 1 is called complexified . A toric arrangement A = { χ − 1 It induces a polyhedral cellularization of ( S 1 ) d : call F ( A ) its face category.
Tools The Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . For J ⊆ I write U J := � i ∈ J U i . U 1 1 U 12 U 13 � � 12 13 23 N ( U ) = 12 13 1 2 3 U 3 U 2 2 23 3 U 23 Nerve of U : the abstract simplicial complex N ( U ) = {∅ � = J ⊆ I | U J � = ∅} Theorem (Weil ‘51, Borsuk ‘48). If U J is contractible for all J ∈ N ( U ), X ≃ | N ( U ) |
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . U 1 D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps.
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . U 1 D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps. X = colim D � � D ( J ) identifying J along maps
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . U 1 D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps. X = colim D hocolim D � � � � ∆ ( n ) × D ( J n ) D ( J ) glue in identifying mapping J 0 ⊆ ... ⊆ J n J along maps cylinders
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . U 1 D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps. X = colim D hocolim D G.N.L.: ≃ � � � � ∆ ( n ) × D ( J n ) D ( J ) glue in identifying mapping J 0 ⊆ ... ⊆ J n J along maps cylinders
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . � U 1 D D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps. hocolim � X = colim D hocolim D D ≃ G.N.L.: ≃ � � � � ∆ ( n ) × D ( J n ) D ( J ) glue in identifying mapping J J 0 ⊆ ... ⊆ J n along maps cylinders
Tools The Generalized Nerve Lemma Let X be a paracompact space with a (locally) finite open cover U = { U i } I . � U 1 D D N ( U ) = � � 1 2 12 U 2 Consider the diagram D : N ( U ) → Top, D ( J ) := U J and inclusion maps. hocolim � X = colim D hocolim D D ≃ G.N.L.: ≃ � � ≃ || N D || � � � � ∆ ( n ) × D ( J n ) D ( J ) glue in identifying mapping J J 0 ⊆ ... ⊆ J n along maps cylinders
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