Alex Suciu Northeastern University Topology Seminar University of - PowerPoint PPT Presentation

D UALITY AND RESONANCE Alex Suciu Northeastern University Topology Seminar University of Rochester November 8, 2017 A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 1 / 28

P OINCARÉ DUALITY P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS Let A be a graded, graded-commutative algebra over a field k . A = À i ě 0 A i , where A i are k -vector spaces. ¨ : A i b A j Ñ A i + j . ab = ( ´ 1 ) ij ba for all a P A i , b P B j . We will assume that A is connected ( A 0 = k ¨ 1), and locally finite (all the Betti numbers b i ( A ) : = dim k A i are finite). A is a Poincaré duality k -algebra of dimension m if there is a k -linear map ε : A m Ñ k (called an orientation ) such that all the bilinear forms A i b k A m ´ i Ñ k , a b b ÞÑ ε ( ab ) are non-singular. Consequently, b i ( A ) = b m ´ i ( A ) , and A i = 0 for i ą m . ε is an isomorphism. The maps PD : A i Ñ ( A m ´ i ) ˚ , PD ( a )( b ) = ε ( ab ) are isomorphisms. Each a P A i has a Poincaré dual , a _ P A m ´ i , such that ε ( aa _ ) = 1. The orientation class is defined as ω A = 1 _ , so that ε ( ω A ) = 1. A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 2 / 28

P OINCARÉ DUALITY T HE ASSOCIATED ALTERNATING FORM T HE ASSOCIATED ALTERNATING FORM Associated to a k - PD m algebra there is an alternating m -form, µ A : Ź m A 1 Ñ k , µ A ( a 1 ^ ¨ ¨ ¨ ^ a m ) = ε ( a 1 ¨ ¨ ¨ a m ) . Assume now that m = 3, and set n = b 1 ( A ) . Fix a basis t e 1 , . . . , e n u for A 1 , and let t e _ 1 , . . . , e _ n u be the PD basis for A 2 . The multiplication in A , then, is given on basis elements by n ÿ µ ijk e _ e i e _ e i e j = k , j = δ ij ω , k = 1 where µ ijk = µ ( e i ^ e j ^ e k ) . Alternatively, let A i = ( A i ) ˚ , and let e i P A 1 be the (Kronecker) dual of e i . We may then view µ dually as a trivector, ÿ µ ijk e i ^ e j ^ e k P Ź 3 A 1 , µ = which encodes the algebra structure of A . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 3 / 28

P OINCARÉ DUALITY P OINCARÉ DUALITY IN ORIENTABLE MANIFOLDS P OINCARÉ DUALITY IN ORIENTABLE MANIFOLDS If M is a compact, connected, orientable, m -dimensional manifold, then the cohomology ring A = H . ( M , k ) is a PD m algebra over k . Sullivan (1975): for every finite-dimensional Q -vector space V and every alternating 3-form µ P Ź 3 V ˚ , there is a closed 3-manifold M with H 1 ( M , Q ) = V and cup-product form µ M = µ . Such a 3-manifold can be constructed via “Borromean surgery." If M bounds an oriented 4-manifold W such that the cup-product pairing on H 2 ( W , M ) is non-degenerate (e.g., if M is the link of an isolated surface singularity), then µ M = 0. A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 4 / 28

R ESONANCE VARIETIES R ESONANCE VARIETIES OF GRADED ALGEBRAS R ESONANCE VARIETIES OF GRADED ALGEBRAS Let A be a connected, finite-type cga over k = C . For each a P A 1 , there is a cochain complex of k -vector spaces, δ 0 δ 1 δ 2 a � A 1 a � A 2 a � ¨ ¨ ¨ , ( A , δ a ) : A 0 with differentials δ a ( b ) = a ¨ b , for b P A i . The resonance varieties of A are the sets s ( A ) = t a P A 1 | dim k H i ( A , δ a ) ě s u . R i An element a P A 1 belongs to R i s ( A ) if and only if rank δ i + 1 + rank δ i a ď b i ( A ) ´ s . a A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 5 / 28

R ESONANCE VARIETIES R ESONANCE VARIETIES OF GRADED ALGEBRAS Fix a k -basis t e 1 , . . . , e n u for A 1 , and let t x 1 , . . . , x n u be the dual basis for A 1 = ( A 1 ) ˚ . Identify Sym ( A 1 ) with S = k [ x 1 , . . . , x n ] , the coordinate ring of the affine space A 1 . Define a cochain complex of free S -modules, L ( A ) : = ( A ‚ b S , δ ) , δ i δ i + 1 � A i + 2 b S � A i b S � A i + 1 b S � ¨ ¨ ¨ , ¨ ¨ ¨ where δ i ( u b s ) = ř n j = 1 e j u b sx j . The specialization of ( A b S , δ ) at a P A 1 coincides with ( A , δ a ) . Hence, R i s ( A ) is the zero-set of the ideal generated by all minors of size b i ´ s + 1 of the block-matrix δ i + 1 ‘ δ i . In particular, R 1 s ( A ) = V ( I n ´ s ( δ 1 )) , the zero-set of the ideal of codimension s minors of δ 1 . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 6 / 28

� � � R ESONANCE VARIETIES R ESONANCE VARIETIES OF GRADED ALGEBRAS R ESONANCE VARIETIES OF PD- ALGEBRAS Let A be a PD m algebra. For all 0 ď i ď m and all a P A 1 , the square ( δ m ´ i ´ 1 ) ˚ � ( A m ´ i ´ 1 ) ˚ a ( A m ´ i ) ˚ PD – PD – δ i A i a A i + 1 commutes up to a sign of ( ´ 1 ) i . Consequently, � ˚ � H i ( A , δ a ) – H m ´ i ( A , δ ´ a ) . Hence, for all i and s , R i s ( A ) = R m ´ i ( A ) . s In particular, R m 1 ( A ) = t 0 u . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 7 / 28

R ESONANCE VARIETIES 3 - DIMENSIONAL P OINCARÉ DUALITY ALGEBRAS 3 - DIMENSIONAL P OINCARÉ DUALITY ALGEBRAS Let A be a PD 3 -algebra with b 1 ( A ) = n ą 0. Then R 3 1 ( A ) = R 0 1 ( A ) = t 0 u . R 2 s ( A ) = R 1 s ( A ) for 1 ď s ď n . R i s ( A ) = H , otherwise. Write R s ( A ) = R 1 s ( A ) . Work of Buchsbaum and Eisenbud on Pfaffians of skew-symmetric matrices implies that R 2 k ( A ) = R 2 k + 1 ( A ) if n is even. R 2 k ´ 1 ( A ) = R 2 k ( A ) if n is odd. If µ A has rank n ě 3, then R n ´ 2 ( A ) = R n ´ 1 ( A ) = R n ( A ) = t 0 u . Here, the rank of a form µ : Ź 3 V Ñ k is the minimum dimension of a linear subspace W Ă V such that µ factors through Ź 3 W . The nullity of µ is the maximum dimension of a subspace U Ă V such that µ ( a ^ b ^ c ) = 0 for all a , b P U and c P V . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 8 / 28

R ESONANCE VARIETIES 3 - DIMENSIONAL P OINCARÉ DUALITY ALGEBRAS If n ě 4, then dim R 1 ( A ) ě null ( µ A ) ě 2. If n is even, then R 1 ( A ) = R 0 ( A ) = A 1 . If n = 2 g + 1 ą 1, then R 1 ( A ) ‰ A 1 if and only if µ A is ‘generic’ in the sense of Berceanu and Papadima (1994). That is, D c P A 1 such that the 2-form γ c P Ź 2 A 1 given by c ‰ 0 in Ź 2 g A 1 . γ c ( a ^ b ) = µ A ( a ^ b ^ c ) has rank 2 g , i.e., γ g In that case, R 1 ( A ) is the hypersurface Pf ( µ A ) = 0, where pf ( δ 1 ( i ; i )) = ( ´ 1 ) i + 1 x i Pf ( µ A ) . E XAMPLE Let M = S 1 ˆ Σ g , where g ě 2. Then µ M = ř g i = 1 a i b i c is generic, and Pf ( µ M ) = x g ´ 1 2 g + 1 . Hence, R 1 = ¨ ¨ ¨ = R 2 g ´ 2 = t x 2 g + 1 = 0 u and R 2 g ´ 1 = R 2 g = R 2 g + 1 = t 0 u . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 9 / 28

R ESONANCE VARIETIES R ESONANCE VARIETIES OF 3 - FORMS OF LOW RANK R ESONANCE VARIETIES OF 3 - FORMS OF LOW RANK n µ R 1 n µ R 1 = R 2 R 3 3 123 0 5 125+345 t x 5 = 0 u 0 R 2 = R 3 n µ R 1 R 4 C 6 6 123+456 t x 1 = x 2 = x 3 = 0 u Y t x 4 = x 5 = x 6 = 0 u 0 C 6 123+236+456 t x 3 = x 5 = x 6 = 0 u 0 n R 1 = R 2 R 3 = R 4 R 5 µ t x 7 = 0 u t x 7 = 0 u 7 147+257+367 0 t x 7 = 0 u t x 4 = x 5 = x 6 = x 7 = 0 u 456+147+257+367 0 123+456+147 t x 1 = 0 u Y t x 4 = 0 u t x 1 = x 2 = x 3 = x 4 = 0 u Y t x 1 = x 4 = x 5 = x 6 = 0 u 0 t x 1 = x 2 = x 4 = x 5 = x 2 123+456+147+257 t x 1 x 4 + x 2 x 5 = 0 u 7 ´ x 3 x 6 = 0 u 0 t x 1 x 4 + x 2 x 5 + x 3 x 6 = x 2 7 u 123+456+147+257+367 0 0 R 2 = R 3 R 4 = R 5 n µ R 1 C 8 8 147+257+367+358 t x 7 = 0 u t x 3 = x 5 = x 7 = x 8 = 0 uYt x 1 = x 3 = x 4 = x 5 = x 7 = 0 u C 8 t x 3 = x 4 = x 5 = x 7 = x 1 x 8 + x 2 456+147+257+367+358 t x 5 = x 7 = 0 u 6 = 0 u C 8 123+456+147+358 t x 1 = x 5 = 0 u Y t x 3 = x 4 = 0 u t x 1 = x 3 = x 4 = x 5 = x 2 x 6 + x 7 x 8 = 0 u C 8 t x 1 = x 5 = 0 u Y t x 3 = x 4 = x 5 = 0 u t x 1 = x 2 = x 3 = x 4 = x 5 = x 7 = 0 u 123+456+147+257+358 C 8 t x 3 = x 5 = x 1 x 4 ´ x 2 7 = 0 u t x 1 = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = 0 u 123+456+147+257+367+358 C 8 t x 1 = x 4 = x 7 = 0 u Y t x 8 = 0 u t x 1 = x 4 = x 7 = x 8 = 0 uYt x 2 = x 3 = x 5 = x 6 = x 8 = 0 u 147+268+358 C 8 147+257+268+358 L 1 Y L 2 Y L 3 L 1 Y L 2 C 8 456+147+257+268+358 C 1 Y C 2 L 1 Y L 2 C 8 L 1 1 Y L 1 2 Y L 1 147+257+367+268+358 L 1 Y L 2 Y L 3 Y L 4 3 C 8 456+147+257+367+268+358 C 1 Y C 2 Y C 3 L 1 Y L 2 Y L 3 C 8 123+456+147+268+358 C 1 Y C 2 L C 8 123+456+147+257+268+358 t f 1 = ¨ ¨ ¨ = f 20 = 0 u 0 123+456+147+257+367+268+358 C 8 t g 1 = ¨ ¨ ¨ = g 20 = 0 u 0 A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 10 / 28

C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite-type CW-complex. The fundamental group π = π 1 ( X , x 0 ) is a finitely presented group, with abelianization π ab – H 1 ( X , Z ) . The group-algebra R = C [ π ab ] is the coordinate ring of the character group, Char ( X ) = Hom ( π , C ˚ ) – ( C ˚ ) n ˆ Tors ( π ab ) , where n = b 1 ( X ) . The characteristic varieties of X are the homology jump loci V i s ( X ) = t ρ P Char ( X ) | dim C H i ( X , C ρ ) ě s u . Away from 1, we have that V 1 s ( X ) = V ( E s ( A π )) , the zero-set of the ideal of codimension s minors of the Alexander matrix of abelianized Fox derivatives of the relators of π . A LEX S UCIU (N ORTHEASTERN ) D UALITY AND RESONANCE R OCHESTER T OP S EMINAR 11 / 28