Abelian Galois covers and rank one local systems Alex Suciu Northeastern University Workshop Université de Nice May 25, 2011 Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 1 / 38
Outline Abelian Galois covers 1 A parameter set for covers The Dwyer–Fried sets Characteristic varieties 2 Jump loci for rank 1 local systems Computing the Ω -invariants Resonance varieties 3 Jump loci for the Aomoto complex Straight spaces Kähler and quasi-Kähler manifolds 4 Jump loci Dwyer–Fried sets Hyperplane arrangements 5 Jump loci and Dwyer–Fried sets Milnor fibration Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 2 / 38
Abelian Galois covers A parameter set for covers Galois covers Sample questions: Given a (finite) CW-complex X , how to parametrize the Galois 1 covers of X with fixed deck-transformation group A ? Given an infinite Galois A -cover, Y → X , are the Betti numbers of 2 Y finite? ◮ If so, how to compute the Betti numbers of Y ? ◮ Furthermore, do the Galois covers of Y have finite Betti numbers? Do the Galois A -covers that have finite Betti numbers form an 3 open subspace of the parameter space? Given a finite Galois A -cover, Y → X , how to compute the Betti 4 numbers of Y ? Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 3 / 38
Abelian Galois covers A parameter set for covers Let X be a connected CW-complex with finite 1-skeleton. We may assume X has a single 0-cell, call it x 0 . Set G = π 1 ( X , x 0 ) . Any epimorphism ν : G ։ A gives rise to a (connected) Galois cover, X ν → X , with group of deck transformations A . Moreover, if α ∈ Aut ( A ) , then X α ◦ ν ∼ = X ν ( A -equivariant homeo). Conversely, if p : ( Y , y 0 ) → ( X , x 0 ) is a Galois A -cover, we get a short exact sequence p ♯ ν � π 1 ( Y , y 0 ) � π 1 ( X , x 0 ) � A � 1 , 1 and an A -equivariant homeomorphism Y ∼ = X ν . Thus, the set of Galois A -covers of X can be identified with Epi ( G , A ) / Aut ( A ) . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 4 / 38
Abelian Galois covers A parameter set for covers Now assume A is a (finitely generated) Abelian group. Then Hom ( G , A ) ← → Hom ( H , A ) , where H = G ab . Proposition (A.S.–Yang–Zhao) There is a bijection Epi ( H , A ) / Aut ( A ) ← → GL n ( Z ) × P Γ where n = rank H, r = rank A, and P is a parabolic subgroup of GL n ( Z ) ; GL n ( Z ) / P = Gr n − r ( Z n ) ; Γ = Epi ( Z n − r ⊕ Tors ( H ) , Tors ( A )) / Aut ( Tors ( A )) —a finite set; GL n ( Z ) × P Γ is the twisted product under the diagonal P -action. Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 5 / 38
� � � Abelian Galois covers A parameter set for covers Simplest situation is when A = Z r . All Galois Z r -covers of X arise as pull-backs of the universal cover of the r -torus: X ν R r f � T r , X where f ♯ : π 1 ( X ) → π 1 ( T r ) realizes the epimorphism ν : G ։ Z r . Hence: � � � � Galois Z r -covers of X r -planes in H 1 ( X , Q ) ← → X ν → X ← → P ν where P ν := im ( ν ∗ : H 1 ( Z r , Q ) → H 1 ( X , Q )) . Thus: Epi ( H , Z r ) / Aut ( Z r ) ∼ = Gr n − r ( Z n ) ∼ = Gr r ( Q n ) . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 6 / 38
Abelian Galois covers The Dwyer–Fried sets The Dwyer–Fried sets Moving about the parameter space for A -covers, and recording how the Betti numbers of those covers vary leads to: Definition The Dwyer–Fried invariants of X are the subsets A ( X ) = { [ ν ] ∈ Epi ( G , A ) / Aut ( A ) | b j ( X ν ) < ∞ , for j ≤ i } . Ω i where X ν → X is the cover corresponding to ν : G ։ A . In particular, when A = Z r , � � � � b j ( X ν ) < ∞ for j ≤ i Ω i P ν ∈ Gr r ( H 1 ( X , Q )) r ( X ) = , with the convention that Ω i r ( X ) = ∅ if r > n = b 1 ( X ) . For a fixed r > 0, get filtration Gr r ( Q n ) = Ω 0 r ( X ) ⊇ Ω 1 r ( X ) ⊇ Ω 2 r ( X ) ⊇ · · · . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 7 / 38
Abelian Galois covers The Dwyer–Fried sets The Ω -sets are homotopy-type invariants: If X ≃ Y , then, for each r > 0, there is an isomorphism Gr r ( H 1 ( Y , Q )) ∼ = Gr r ( H 1 ( X , Q )) sending each subset Ω i r ( Y ) bijectively onto Ω i r ( X ) . Thus, we may extend the definition of the Ω -sets from spaces to groups: Ω i r ( G ) = Ω i r ( K ( G , 1 )) , and similarly for Ω i A ( X ) . Example Let X = S 1 ∨ S k , for some k > 1. Then X ab ≃ � j ∈ Z S k j . Thus, � { pt } for i < k , Ω i 1 ( X ) = ∅ for i ≥ k . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 8 / 38
� � � Abelian Galois covers The Dwyer–Fried sets Comparison diagram There is an commutative diagram, Epi ( G , A ) / Aut A ∼ Ω i = GL n ( Z ) × P Γ A ( X ) � � Ω i � Gr r ( Q n ) r ( X ) � � If Ω i r ( X ) = ∅ , then Ω i A ( X ) = ∅ . The above is a pull-back diagram if and only if: If X ν is a Z r -cover with finite Betti numbers up to degree i , then any regular Tors ( A ) -cover of X ν has the same finiteness property. Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 9 / 38
Abelian Galois covers The Dwyer–Fried sets Example Let X = S 1 ∨ RP 2 . Then G = Z ∗ Z 2 , G ab = Z ⊕ Z 2 , G fab = Z , and � � � X fab ≃ X ab ≃ RP 2 S 1 S 2 j , j ∨ j . j ∈ Z j ∈ Z j ∈ Z Thus, b 1 ( X fab ) = 0, yet b 1 ( X ab ) = ∞ . Hence, Ω 1 1 ( X ) � = ∅ , but Ω 1 Z ⊕ Z 2 ( X ) = ∅ . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 10 / 38
Characteristic varieties Jump loci for rank 1 local systems Characteristic varieties Group of complex-valued characters of G : � G = Hom ( G , C × ) = H 1 ( X , C × ) Let G ab = G / G ′ ∼ = H 1 ( X , Z ) be the abelianization of G . The map ≃ ab : G ։ G ab induces an isomorphism � → � G ab − G . G 0 = ( C × ) n , an algebraic torus of dimension n = rank G ab . � G = � � Tors ( G ab ) ( C × ) n . � G parametrizes rank 1 local systems on X : ρ : G → C × C ρ � the complex vector space C , viewed as a right module over the group ring Z G via a · g = ρ ( g ) a , for g ∈ G and a ∈ C . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 11 / 38
Characteristic varieties Jump loci for rank 1 local systems The homology groups of X with coefficients in C ρ are defined as H ∗ ( X , C ρ ) = H ∗ ( C ρ ⊗ Z G C • ( � X , Z )) , where C • ( � X , Z ) is the Z G -equivariant cellular chain complex of the universal cover of X . Definition The characteristic varieties of X are the sets V i ( X ) = { ρ ∈ � G | H j ( X , C ρ ) � = 0 , for some j ≤ i } . Get filtration { 1 } = V 0 ( X ) ⊆ V 1 ( X ) ⊆ · · · ⊆ � G . If X has finite k -skeleton, then V i ( X ) is a Zariski closed subset of the algebraic group � G , for each i ≤ k . The varieties V i ( X ) are homotopy-type invariants of X . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 12 / 38
Characteristic varieties Jump loci for rank 1 local systems The characteristic varieties may be reinterpreted as the support varieties for the Alexander invariants of X . Let X ab → X be the maximal abelian cover. View H ∗ ( X ab , C ) as a module over C [ G ab ] . Then � � � �� � � V i ( X ) = V X ab , C ann H j . j ≤ i Let X fab → X be the max free abelian cover. View H ∗ ( X fab , C ) as a module over C [ G fab ] ∼ = Z [ t ± 1 1 , . . . , t ± 1 n ] , where n = b 1 ( G ) . Then � � � �� � � G 0 = V W i ( X ) := V i ( X ) ∩ � X fab , C ann H j . j ≤ i Example Let L = ( L 1 , . . . , L n ) be a link in S 3 , with complement X = S 3 \ � n i = 1 L i and Alexander polynomial ∆ L = ∆ L ( t 1 , . . . , t n ) . Then V 1 ( X ) = { z ∈ ( C × ) n | ∆ L ( z ) = 0 } ∪ { 1 } . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 13 / 38
Characteristic varieties Jump loci for rank 1 local systems The characteristic varieties V i j ( X , k ) = { ρ ∈ Hom ( π 1 ( X ) , k × ) | dim k H i ( X , k ρ ) ≥ j } can be used to compute the homology of finite abelian Galois covers (work of A. Libgober, E. Hironaka, P . Sarnak–S. Adams, M. Sakuma, D. Matei–A. S. from the 1990s). E.g.: Theorem (Matei–A.S. 2002) Let ν : π 1 ( X ) ։ Z n . Suppose ¯ k = k and char k ∤ n, so that Z n ⊂ k × . Then: � dim k H 1 ( X ν , k ) = dim k H 1 ( X , k ) + ϕ ( k ) · depth k ( ν n / k ) , 1 � = k | n where depth k ( ρ ) = max { j | ρ ∈ V 1 j ( X , k ) } . Alex Suciu (Northeastern Univ.) Abelian Galois covers and local systems Univ. de Nice, May 25, 2011 14 / 38
Recommend
More recommend
