G EOMETRY AND TOPOLOGY OF COHOMOLOGY JUMP LOCI L ECTURE 1: C HARACTERISTIC VARIETIES Alex Suciu Northeastern University MIMS Summer School: New Trends in Topology and Geometry Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia July 9–12, 2018 A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 1 / 25
O UTLINE C AST OF CHARACTERS 1 The character group The equivariant chain complex Characteristic varieties Degree 1 characteristic varieties E XAMPLES AND COMPUTATIONS 2 Warm-up examples Toric complexes and RAAGs Quasi-projective manifolds A PPLICATIONS 3 Homology of finite abelian covers Dwyer–Fried sets Duality and propagation A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 2 / 25
C AST OF CHARACTERS T HE CHARACTER GROUP T HE CHARACTER GROUP Throughout, X will be a connected CW-complex with finite q -skeleton, for some q ě 1. We may assume X has a single 0-cell, call it e 0 . Let G = π 1 ( X , e 0 ) be the fundamental group of X : a finitely generated group, with generators x 1 = [ e 1 1 ] , . . . , x m = [ e 1 m ] . The character group , p G = Hom ( G , C ˆ ) Ă ( C ˆ ) m is a (commutative) algebraic group, with multiplication ρ ¨ ρ 1 ( g ) = ρ ( g ) ρ 1 ( g ) , and identity G Ñ C ˆ , g ÞÑ 1. Let G ab = G / G 1 – H 1 ( X , Z ) be the abelianization of G . The projection ab : G Ñ G ab induces an isomorphism p » Ñ p G ab Ý G . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 3 / 25
C AST OF CHARACTERS T HE CHARACTER GROUP The identity component, p G 0 , is isomorphic to a complex algebraic torus of dimension n = rank G ab . The other connected components are all isomorphic to G 0 = ( C ˆ ) n , and are indexed by the finite abelian group p Tors ( G ab ) . Char ( X ) = p G is the moduli space of rank 1 local systems on X : ρ : G Ñ C ˆ 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 P G and a P C . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 4 / 25
C AST OF CHARACTERS T HE EQUIVARIANT CHAIN COMPLEX T HE EQUIVARIANT CHAIN COMPLEX Let p : r X Ñ X be the universal cover. The cell structure on X lifts to a cell structure on r X . e 0 P p ´ 1 ( e 0 ) identifies G = π 1 ( X , e 0 ) with the group of Fixing a lift ˜ deck transformations of r X . Thus, we may view the cellular chain complex of r X as a chain complex of left Z G -modules, ˜ ˜ B i + 1 � C i ( r B i � C i + 1 ( r � C i ´ 1 ( r � ¨ ¨ ¨ . ¨ ¨ ¨ X , Z ) X , Z ) X , Z ) ˜ e 1 e 0 . B 1 ( ˜ i ) = ( x i ´ 1 ) ˜ e 2 ) = ř m � φ ¨ ˜ ˜ e 1 � B 2 ( ˜ B r / B x i i , where i = 1 r P F m = x x 1 , . . . , x m y is the word traced by the attaching map of e 2 ; B r / B x i P Z F m are the Fox derivatives of r ; φ : Z F m Ñ Z G is the linear extension of the projection F m ։ G . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 5 / 25
C AST OF CHARACTERS T HE EQUIVARIANT CHAIN COMPLEX H ˚ ( X , C ρ ) is the homology of the chain complex of C -vector spaces C ρ b Z G C ‚ ( r X , Z ) : ˜ ˜ B i + 1 ( ρ ) � C i ( X , C ) B i ( ρ ) � C i + 1 ( X , C ) � C i ´ 1 ( X , C ) � ¨ ¨ ¨ , ¨ ¨ ¨ where the evaluation of ˜ B i at ρ is obtained by applying the ring homomorphism Z G Ñ C , g ÞÑ ρ ( g ) to each entry of ˜ B i . Alternatively, consider the universal abelian cover, X ab , and its equivariant chain complex, C ‚ ( X ab , Z ) = Z G ab b Z G C ‚ ( r X , Z ) , = id b r with differentials B ab B i . i Then H ˚ ( X , C ρ ) is computed from the resulting C -chain complex, i ( ρ ) = ˜ with differentials B ab B i ( ρ ) . The identity 1 P Char ( X ) yields the trivial local system, C 1 = C , and H ˚ ( X , C ) is the usual homology of X with C -coefficients. Denote by b i ( X ) = dim C H i ( X , C ) the i th Betti number of X . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 6 / 25
C AST OF CHARACTERS C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES D EFINITION The characteristic varieties of X are the sets V i k ( X ) = t ρ P Char ( X ) | dim C H i ( X , C ρ ) ě k u . For each i , get stratification Char ( X ) = V i 0 Ě V i 1 Ě V i 2 Ě ¨ ¨ ¨ 1 P V i k ( X ) ð ñ b i ( X ) ě k . V 0 1 ( X ) = t 1 u and V 0 k ( X ) = H , for k ą 1. Define analogously V i k ( X , k ) Ă Hom ( G , k ˆ ) , for arbitrary field k . Then V i k ( X , k ) = V i k ( X , K ) X Hom ( G , k ˆ ) , for any k Ď K . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 7 / 25
C AST OF CHARACTERS C HARACTERISTIC VARIETIES L EMMA For each 0 ď i ď q and k ě 0 , the set V i k ( X ) is a Zariski closed subset of the algebraic group p G = Char ( X ) . P ROOF ( FOR i ă q ). Let R = C [ G ab ] be the coordinate ring of p G = p G ab . By definition, a character ρ belongs to V i k ( X ) if and only if rank B ab i + 1 ( ρ ) + rank B ab i ( ρ ) ď c i ´ k , where c i = c i ( X ) is the number of i -cells of X . Hence, č t ρ P p V i G | rank B ab i + 1 ( ρ ) ď r ´ 1 or rank B ab k ( X ) = i ( ρ ) ď s ´ 1 u r + s = c i ´ k + 1 ; r , s ě 0 ÿ � � I r ( B ab i ) ¨ I s ( B ab = V i + 1 ) , r + s = c i ´ k + 1 ; r , s ě 0 where I r ( ϕ ) = ideal of r ˆ r minors of ϕ . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 8 / 25
C AST OF CHARACTERS C HARACTERISTIC VARIETIES The characteristic varieties are homotopy-type invariants of a space: L EMMA G 1 – p Suppose X » X 1 . There is then an isomorphism p G, which restricts to isomorphisms V i k ( X 1 ) – V i k ( X ) , for all i ď q and k ě 0 . P ROOF . Let f : X Ñ X 1 be a (cellular) homotopy equivalence. 1 0 ) , yields an The induced homomorphism f 7 : π 1 ( X , e 0 ) Ñ π 1 ( X 1 , e f 7 : x G 1 Ñ p isomorphism of algebraic groups, ˆ G . Lifting f to a cellular homotopy equivalence, ˜ f : r X Ñ r X 1 , defines isomorphisms H i ( X , C ρ ˝ f 7 ) Ñ H i ( X 1 , C ρ ) , for each ρ P p G 1 . Hence, ˆ f 7 restricts to isomorphisms V i k ( X 1 ) Ñ V i k ( X ) . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 9 / 25
D EGREE 1 CHARACTERISTIC VARIETIES C AST OF CHARACTERS D EGREE 1 CHARACTERISTIC VARIETIES V 1 k ( X ) depends only on G = π 1 ( X ) (in fact, only on G / G 2 ), so we may write these sets as V 1 k ( G ) . Suppose G = x x 1 , . . . , x m | r 1 , . . . , r p y is finitely presented Away from 1 P p G , we have that V 1 k ( G ) = V ( E k ( B ab 1 )) , the zero-set of the ideal of codimension k minors of the Alexander matrix � ab : Z G p B ab ab Ñ Z G m � 1 = B r i / B x j ab . If ϕ : G ։ Q is an epimorphism, then, for each k ě 1, the induced monomorphism between character groups, ϕ ˚ : p Ñ p Q ã G , restricts to an embedding V 1 Ñ V 1 k ( Q ) ã k ( G ) . Given any subvariety W Ă ( C ˆ ) n defined over Z , there is a finitely presented group G such that G ab = Z n and V 1 1 ( G ) = W . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 10 / 25
E XAMPLES AND COMPUTATIONS W ARM - UP EXAMPLES W ARM - UP EXAMPLES E XAMPLE (T HE CIRCLE ) We have Ă S 1 = R . Identify π 1 ( S 1 , ˚ ) = Z = x t y and ZZ = Z [ t ˘ 1 ] . Then: C ‚ ( Ă t ´ 1 � Z [ t ˘ 1 ] � Z [ t ˘ 1 ] � 0 S 1 ) : 0 For ρ P Hom ( Z , C ˆ ) = C ˆ , we get ρ ´ 1 � C C ρ b ZZ C ‚ ( Ă � C � 0 S 1 ) : 0 which is exact, except for ρ = 1, when H 0 ( S 1 , C ) = H 1 ( S 1 , C ) = C . Hence: V 0 1 ( S 1 ) = V 1 1 ( S 1 ) = t 1 u V i k ( S 1 ) = H , otherwise. A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 11 / 25
E XAMPLES AND COMPUTATIONS W ARM - UP EXAMPLES E XAMPLE (T HE n - TORUS ) Identify π 1 ( T n ) = Z n , and Hom ( Z n , C ˆ ) = ( C ˆ ) n . Using the Koszul resolution C ‚ ( Ă T n ) as above, we get # if k ď ( n t 1 u i ) , V i k ( T n ) = H otherwise . E XAMPLE (N ILMANIFOLDS ) More generally, let M be a nilmanifold. An inductive argument on the nilpotency class of π 1 ( M ) , based on the Hochschild-Serre spectral sequence, yields # t 1 u if k ď b i ( M ) , V i k ( M ) = H otherwise A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 12 / 25
E XAMPLES AND COMPUTATIONS W ARM - UP EXAMPLES E XAMPLE (W EDGE OF CIRCLES ) Identify π 1 ( Ž n S 1 ) = F n , and Hom ( F n , C ˆ ) = ( C ˆ ) n . Then: $ ( C ˆ ) n ’ if k ă n , & ł n S 1 � = V 1 � t 1 u if k = n , k ’ % H if k ą n . E XAMPLE (O RIENTABLE SURFACE OF GENUS g ą 1 ) Write π 1 ( Σ g ) = x x 1 , . . . , x g , y 1 , . . . , y g | [ x 1 , y 1 ] ¨ ¨ ¨ [ x g , y g ] = 1 y , and identify Hom ( π 1 ( Σ g ) , C ˆ ) = ( C ˆ ) 2 g . Then: $ ( C ˆ ) 2 g ’ if i = 1, k ă 2 g ´ 1 , & V i k ( Σ g ) = t 1 u if i = 1, k = 2 g ´ 1 , 2 g ; or i = 2, k = 1 , ’ % H otherwise . A LEX S UCIU (N ORTHEASTERN ) C OHOMOLOGY JUMP LOCI MIMS S UMMER S CHOOL 2018 13 / 25
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