Combinatorics and topology of toric arrangements II. Topology of - PowerPoint PPT Presentation

Combinatorics and topology of toric arrangements II. Topology of arrangements in the complex torus (An invitation to combinatorial algebraic topology) Emanuele Delucchi (SNSF / Universit´ e de Fribourg) Toblach/Dobbiaco February 23, 2017

Toric arrangements Recall: a toric arrangement in the complex torus T := ( C ⇤ ) d is a set A := { K 1 , . . . , K n } of ‘hypertori’ K i = { z 2 T | z a i = b i } with a i 2 Z d \ 0 and b i 2 C ⇤ The complement of A is M ( A ) := T \ [ A , Problem: Study the topology of M ( A ).

The long game Let A = [ a 1 , . . . , a n ] 2 M d ⇥ n ( Z ) (Central) hyperplane (Centered) toric (Centered) elliptic arrangement arrangement arrangement C d ! C ( C ⇤ ) d ! C ⇤ E d ! E � i : � i : � i : z 7! P z 7! Q z 7! P j z a ji j a ji z j j a ji z j j H i := ker � i K i := ker � i L i := ker � i A = { H 1 , . . . , H n } A = { K 1 , . . . , K n } A = { L 1 , . . . , L n } M ( A ) := C d \ [ A M ( A ) := ( C ⇤ ) d \ [ A M ( A ) := E d \ [ A C ( A ) rk : 2 [ n ] ! N ? m : 2 [ n ] ! N M ( A )

Context Hyperplanes: Brieskorn A := { H 1 , . . . , H d } : set of (a ffi ne) hyperplanes in C d , C ( A ) = L ( A ) := { \ B | B ✓ A } : poset of intersections (reverse inclusion). For X 2 L ( A ): A X = { H i 2 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 M ⇠ = H k ( M ( A X ) , Z ) ! H k ( M ( A ) , Z ) b : � X 2 L ( A ) codim X = k

Context Hyperplanes: Brieskorn A := { H 1 , . . . , H d } : set of (a ffi ne) hyperplanes in C d , C ( A ) = L ( A ) := { \ B | B ✓ A } : poset of intersections (reverse inclusion). For X 2 L ( A ): A X = { H i 2 A | X ✓ H i } . A L ( A ) A X X In fact: M ( A ) is a minimal space , i.e., it has the homotopy type of a CW- complex with as many cells in dimension k as there are generators in k -th cohomology. [Dimca-Papadima ‘03]

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 h P k l =1 ( � 1) l e j 1 · · · c e j l · · · e j k | codim( \ i =1 ...k H j i ) = k � 1 i L ( A ) Fully determined by L ( A ) (cryptomorphisms!). codim X For instance: X µ L ( A ) (ˆ ( � t ) rk X P ( M ( A ) , t ) = 0 , X ) | {z } X 2 L ( A ) M¨ obius Poin( M ( A ) , t ) = function of L ( A ) 1 + 4 t + 5 t 2 + 2 t 3

Context Toric arrangements Another good reason for considering C ( A ), the poset of layers (i.e. con- nected components of intersections of the K i ). A : C ( A ): Theorem [Looijenga ‘93, De Concini-Procesi ‘05] X µ C ( A ) ( Y )( � t ) rk Y (1 + t ) d � rk Y Poin( M ( A ) , Z ) = Y 2 C ( A ) = ( � t ) d � C ( A ) ( � t (1 + t ))

Context Toric arrangements [De Concini – Procesi ’05] compute the Poincar´ e polynomial and the cup product in H ⇤ ( M ( A ) , C ) when the matrix [ a 1 , . . . , a n ] is totally unimodular. [d’Antonio–D. ‘11,‘13] ⇡ 1 ( M ( A )), minimality, torsion-freeness (complexified) [Bibby ’14] Q -cohomology algebra of unimodular abelian arrangements [Dupont ’14] Algebraic model for C -cohomology algebra of complements of hypersurface arrangements in manifolds with hyperplane-like crossings. [Callegaro-D. ‘15] Integer cohomology algebra, its dependency from C ( A ). [Bergvall ‘16] Cohomology as repr. of Weyl group in type G 2 , F 4 , E 6 , E 7 . Wonderful models: nonprojective [Moci‘12] , projective [Gai ffi -De Concini ‘16] .

Tools Posets and categories P - a partially ordered set C - a s.c.w.o.l. (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 8 9 a b c < = ( ∅ ) ab ac b c : ; || P || P ∆ P C

Tools Posets and categories P - a partially ordered set C - a s.c.w.o.l. (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] Example 1: X regular: F ( X ) is a poset, ||F ( X ) || = Bd( X ). Example 2: A complexified toric arrangement ( A = { � � 1 i ( b i ) } with b i 2 S 1 ) 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 := T i 2 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 ) = { ; 6 = J ✓ I | U J 6 = ; } Theorem (Weil ‘51, Borsuk ‘48). If U J is contractible for all J 2 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 . b 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 b X = colim D hocolim D D ' G.N.L.: ' R b ' || N D || � � ] ] ∆ ( n ) ⇥ D ( J n ) Grothendieck D ( J ) glue in identifying construction 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 . b 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 b X = colim D hocolim D D ' G.N.L.: ' R b ' || N D || � � ] ] ∆ ( n ) ⇥ D ( J n ) Grothendieck D ( J ) glue in identifying construction mapping J J 0 ✓ ... ✓ J n (...whatever.) along maps cylinders

Tools The Generalized Nerve Lemma Application: the Salvetti complex Let A be a complexified arrangement of hyperplanes in C d (i.e. the defining equations for the hyperplanes are real). [Salvetti ‘87] There is a poset Sal( A ) such that || Sal( A ) || ' M ( A ) . Recall: complexified means ↵ i 2 ( R d ) ⇤ and b i 2 R . Consider the associated arrangement A R = { H R i } in R d , H R i = < ( H i ).

The Salvetti poset For z 2 C d and all j , ↵ j ( z ) = ↵ j ( < ( z )) + i ↵ j ( = ( z )). We have z 2 M ( A ) if and only if ↵ j ( z ) 6 = 0 for all j . Thus, surely for very region (chamber) C 2 R ( A R ) we have U ( C ) := C + i R d ✓ M ( A ) . n o G.N.L. applies to the covering by M ( A )-closed sets U := U ( C ) C 2 R ( A R ) (what’s important is that each ( M ( A ) , U ( C )) is NDR-pair). R b After some “massaging”, N ( U ) D becomes 2 F 2 F ( A R ) , C 2 R ( A R { [ F, C ] | | F | ) } , 6 | {z } 6 Sal( A ) = 4 $ C 2 R ( A R ) ,C  F [ F, C ] � [ F 0 , C 0 ] if F  F 0 , C ✓ C 0

Salvetti complexes of pseudoarrangements Notice: the definition of Sal( A ) makes sense also for general pseudoarrange- ments (oriented matroids). Theorem. [D.–Falk ‘15] The class of complexes || Sal( A ) || where A is a pseu- doarrangement gives rise to “new” fundamental groups. For instance, the non-pappus oriented matroid gives rise to a fundamental group that is not isomorphic to any realizable arrangement group.

Tools The Generalized Nerve Lemma Application: the Salvetti complex Let A be a complexified arrangement of hyperplanes in C d (i.e. the defining equations for the hyperplanes are real). [Salvetti ‘87] There is a poset Sal( A ) such that || Sal( A ) || ' M ( A ) . [Callegaro-D. ‘15] Let X 2 L ( A ) with codim X = k . There is a map of posets Sal( A ) ! Sal( A X ) that induces the Brieskorn inclusion b X : H k ( M ( A X ) , Z ) , ! H k ( M ( A ) , Z ). Q: ”Brieskorn decomposition” in the (“wiggly”) case of oriented matroids?

Salvetti Category [d’Antonio-D., ‘11] Any complexified toric arrangement A lifts to a complexified arrangement of a ffi ne hyperplanes A � under the universal cover / Z d C d ! T, A � : � ! A : The group Z d acts on Sal( A � ) and we can define the Salvetti category of A : Sal( A ) := Sal( A � ) / Z d (quotient taken in the category of scwols). Here the realization commutes with the quotient [Babson-Kozlov ‘07] , thus || Sal( A ) || ' M ( A ) .

Tools Discrete Morse Theory [Forman, Chari, Kozlov,...; since ’98] Here is a regular CW complex with its poset of cells:

Tools Discrete Morse Theory [Forman, Chari, Kozlov,...; since ’98] Elementary collapses... ... are homotopy equivalences.

Tools Discrete Morse Theory [Forman, Chari, Kozlov,...; since ’98] Elementary collapses... ... are homotopy equivalences.

Tools Discrete Morse Theory [Forman, Chari, Kozlov,...; since ’98] Elementary collapses... ... are homotopy equivalences.