A LGEBRAIC MODELS , DUALITY , AND RESONANCE Alex Suciu Northeastern University Topology Seminar MIT March 5, 2018 A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 1 / 24
D UALITY PROPERTIES P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY ALGEBRAS Let A be a graded, graded-commutative algebra over a field k . i ě 0 A i , where A i are k -vector spaces. A = À ¨ : 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 n if there is a k -linear map ε : A n Ñ k (called an orientation ) such that all the bilinear forms A i b k A n ´ i Ñ k , a b b ÞÑ ε ( ab ) are non-singular. Consequently, b i ( A ) = b n ´ i ( A ) , and A i = 0 for i ą n . ε is an isomorphism. The maps PD : A i Ñ ( A n ´ i ) ˚ , PD ( a )( b ) = ε ( ab ) are isomorphisms. Each a P A i has a Poincaré dual , a _ P A n ´ 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 ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 2 / 24
D UALITY PROPERTIES P OINCARÉ DUALITY ALGEBRAS T HE ASSOCIATED ALTERNATING FORM Associated to a k - PD n algebra there is an alternating n -form, µ A : Ź n A 1 Ñ k , µ A ( a 1 ^ ¨ ¨ ¨ ^ a n ) = ε ( a 1 ¨ ¨ ¨ a n ) . Assume now that n = 3, and set r = b 1 ( A ) . Fix a basis t e 1 , . . . , e r u for A 1 , and let t e _ 1 , . . . , e _ r u be the dual basis for A 2 . The multiplication in A , then, is given on basis elements by r 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, µ ijk e i ^ e j ^ e k P Ź 3 A 1 , ÿ µ = which encodes the algebra structure of A . A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 3 / 24
D UALITY PROPERTIES P OINCARÉ DUALITY ALGEBRAS P OINCARÉ DUALITY IN ORIENTABLE MANIFOLDS If M is a compact, connected, orientable, n -dimensional manifold, then the cohomology ring A = H . ( M , k ) is a PD n 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 ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 4 / 24
D UALITY PROPERTIES D UALITY SPACES D UALITY SPACES A more general notion of duality is due to Bieri and Eckmann (1978). Let X be a connected, finite-type CW-complex, and set π = π 1 ( X , x 0 ) . X is a duality space of dimension n if H i ( X , Z π ) = 0 for i ‰ n and H n ( X , Z π ) ‰ 0 and torsion-free. Let D = H n ( X , Z π ) be the dualizing Z π -module. Given any Z π -module A , we have H i ( X , A ) – H n ´ i ( X , D b A ) . If D = Z , with trivial Z π -action, then X is a Poincaré duality space. If X = K ( π , 1 ) is a duality space, then π is a duality group . A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 5 / 24
D UALITY PROPERTIES A BELIAN DUALITY SPACES A BELIAN DUALITY SPACES We introduce in [Denham–S.–Yuzvinsky 2016/17] an analogous notion, by replacing π � π ab . X is an abelian duality space of dimension n if H i ( X , Z π ab ) = 0 for i ‰ n and H n ( X , Z π ab ) ‰ 0 and torsion-free. Let B = H n ( X , Z π ab ) be the dualizing Z π ab -module. Given any Z π ab -module A , we have H i ( X , A ) – H n ´ i ( X , B b A ) . The two notions of duality are independent: E XAMPLE Surface groups of genus at least 2 are not abelian duality groups, though they are (Poincaré) duality groups. Let π = Z 2 ˚ G , where G = x x 1 , . . . , x 4 | x ´ 2 1 x 2 x 1 x ´ 1 2 , . . . , x ´ 2 4 x 1 x 4 x ´ 1 1 y is Higman’s acyclic group. Then π is an abelian duality group (of dimension 2), but not a duality group. A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 6 / 24
D UALITY PROPERTIES A RRANGEMENTS OF SMOOTH HYPERSURFACES T HEOREM (DSY) Let X be an abelian duality space of dimension n. Then: b 1 ( X ) ě n ´ 1 . b i ( X ) ‰ 0 , for 0 ď i ď n and b i ( X ) = 0 for i ą n. ( ´ 1 ) n χ ( X ) ě 0 . T HEOREM (D ENHAM –S. 2017) Let U be a connected, smooth, complex quasi-projective variety of dimension n. Suppose U has a smooth compactification Y for which Components of Y z U form an arrangement of hypersurfaces A ; 1 For each submanifold X in the intersection poset L ( A ) , the 2 complement of the restriction of A to X is a Stein manifold. Then U is both a duality space and an abelian duality space of dimension n. A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 7 / 24
D UALITY PROPERTIES A RRANGEMENTS OF SMOOTH HYPERSURFACES L INEAR , ELLIPTIC , AND TORIC ARRANGEMENTS T HEOREM (DS17) Suppose that A is one of the following: An affine-linear arrangement in C n , or a hyperplane arrangement 1 in CP n ; A non-empty elliptic arrangement in E n ; 2 A toric arrangement in ( C ˚ ) n . 3 Then the complement M ( A ) is both a duality space and an abelian duality space of dimension n ´ r, n + r, and n, respectively, where r is the corank of the arrangement. This theorem extends several previous results: Davis, Januszkiewicz, Leary, and Okun (2011); 1 Levin and Varchenko (2012); 2 Davis and Settepanella (2013), Esterov and Takeuchi (2014). 3 A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 8 / 24
A LGEBRAIC MODELS AND RESONANCE VARIETIES C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS Let A = ( A ‚ , d ) be a commutative, differential graded algebra over a field k of characteristic 0. That is: i ě 0 A i , where A i are k -vector spaces. A = À The multiplication ¨ : A i b A j Ñ A i + j is graded-commutative, i.e., ab = ( ´ 1 ) | a || b | ba for all homogeneous a and b . The differential d : A i Ñ A i + 1 satisfies the graded Leibnitz rule, i.e., d ( ab ) = d ( a ) b + ( ´ 1 ) | a | a d ( b ) . A CDGA A is of finite-type (or q-finite ) if it is connected (i.e., A 0 = k ¨ 1) and dim A i ă 8 for all i ď q . H ‚ ( A ) inherits an algebra structure from A . A cdga morphism ϕ : A Ñ B is both an algebra map and a cochain map. Hence, it induces a morphism ϕ ˚ : H ‚ ( A ) Ñ H ‚ ( B ) . A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 9 / 24
A LGEBRAIC MODELS AND RESONANCE VARIETIES C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS A map ϕ : A Ñ B is a quasi-isomorphism if ϕ ˚ is an isomorphism. Likewise, ϕ is a q -quasi-isomorphism (for some q ě 1) if ϕ ˚ is an isomorphism in degrees ď q and is injective in degree q + 1. Two cdgas, A and B , are (q-)equivalent ( » q ) if there is a zig-zag of ( q -)quasi-isomorphisms connecting A to B . A cdga A is formal (or just q-formal ) if it is ( q -)equivalent to ( H ‚ ( A ) , d = 0 ) . A CDGA is q-minimal if it is of the form ( Ź V , d ) , where the differential structure is the inductive limit of a sequence of Hirsch extensions of increasing degrees, and V i = 0 for i ą q . Every CDGA A with H 0 ( A ) = k admits a q-minimal model , M q ( A ) (i.e., a q -equivalence M q ( A ) Ñ A with M q ( A ) = ( Ź V , d ) a q -minimal cdga), unique up to iso. A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 10 / 24
A LGEBRAIC MODELS AND RESONANCE VARIETIES A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES Given any (path-connected) space X , there is an associated Sullivan Q -cdga, A PL ( X ) , such that H ‚ ( A PL ( X )) = H ‚ ( X , Q ) . An algebraic (q-)model (over k ) for X is a k -cgda ( A , d ) which is ( q -) equivalent to A PL ( X ) b Q k . If M is a smooth manifold, then Ω dR ( M ) is a model for M (over R ). Examples of spaces having finite-type models include: Formal spaces (such as compact Kähler manifolds, hyperplane arrangement complements, toric spaces, etc). Smooth quasi-projective varieties, compact solvmanifolds, Sasakian manifolds, etc. A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 11 / 24
A LGEBRAIC MODELS AND RESONANCE VARIETIES R ESONANCE VARIETIES OF A CDGA R ESONANCE VARIETIES OF A CDGA Let A = ( A ‚ , d ) be a connected, finite-type CDGA over k = C . For each a P Z 1 ( A ) – H 1 ( A ) , we have a cochain complex, δ 0 δ 1 δ 2 a � A 1 a � A 2 a � ¨ ¨ ¨ , ( A ‚ , δ a ) : A 0 with differentials δ i a ( u ) = a ¨ u + d u , for all u P A i . The resonance varieties of A are the affine varieties R i s ( A ) = t a P H 1 ( A ) | dim H i ( A ‚ , δ a ) ě s u . If A is a CGA (that is, d = 0), the resonance varieties R i s ( A ) are homogeneous subvarieties of A 1 . If X is a connected, finite-type CW-complex, we set R i s ( X ) : = R i s ( H ‚ ( X , C )) . A LEX S UCIU (N ORTHEASTERN ) M ODELS , DUALITY , AND RESONANCE MIT T OPOLOGY S EMINAR 12 / 24
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