F ORMALITY NOTIONS FOR SPACES AND GROUPS Alex Suciu Northeastern University Workshop on Nilpotent Fundamental Groups Banff International Research Station June 22, 2017 A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 1 / 24
A LGEBRAIC MODELS AND FORMALITY C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS Let A “ p A ‚ , d q be a commutative, differential graded algebra over a field k of characteristic 0. A “ À i ě 0 A i , where A i are k -vector spaces. The multiplication ¨ : A i b A j Ñ A i ` j is graded-commutative, i.e., ab “ p´ 1 q | 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 p ab q “ d p a q b ` p´ 1 q | a | a d p b q . The cohomology H ‚ p A q of the cochain complex p A , d q inherits an algebra structure from A . A cdga morphism ϕ : A Ñ B is both an algebra map and a cochain map. Hence, ϕ induces a morphism ϕ ˚ : H ‚ p A q Ñ H ‚ p B q . The map ϕ 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. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 2 / 24
A LGEBRAIC MODELS AND FORMALITY F ORMALITY OF CDGA S F ORMALITY OF CDGA S Two cdgas, A and B , are weakly (q-)equivalent ( » q ) if there is a zig-zag of ( q -)quasi-isomorphisms connecting A to B . (Sullivan 1977) A cdga p A , d q is formal (or just q-formal ) if it is ( q -)weakly equivalent to p H ‚ p A q , d “ 0 q . Formality implies uniform vanishing of all Massey products. E.g., if A is 1-formal, then all triple Massey products in H 2 p A q must vanish modulo indeterminancy: if a , b , c P H 1 p A q , and ab “ bc “ 0, then x a , b , c y “ 0 in H ‚ p A q{p a , c q . (Halperin–Stasheff 1979) Let K { k be a field extension. A k -cdga p A , d q with H ‚ p A q of finite-type is formal if and only if the K -cdga p A b K , d b id K q is formal. (S.–He Wang 2015) Suppose dim H ď q ` 1 p A q ă 8 and H 0 p A q “ k . Then p A , d q is q -formal iff p A b K , d b id K q is q -formal. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 3 / 24
A LGEBRAIC MODELS AND FORMALITY A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES To a large extent, the rational homotopy type of a space can be reconstructed from algebraic models associated to it. If the space is a smooth manifold M , the standard R -model is the de Rham algebra Ω dR p M q . More generally, any (path-connected) space X has an associated Sullivan Q -cdga, A PL p X q . In particular, H ‚ p A PL p X qq “ H ‚ p X , Q q . An algebraic (q-)model (over k ) for X is a k -cgda p A , d q which is ( q -) weakly equivalent to A PL p X q b Q k . For instance, every smooth, quasi-projective variety X admits a finite-dimensional, rational model A “ A p X , D q , constructed by Morgan from a normal-crossings compactification X “ X Y D . A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 4 / 24
A LGEBRAIC MODELS AND FORMALITY F ORMALITY OF SPACES F ORMALITY OF SPACES A space X is (q-)formal if A PL p X q has this property, i.e., p H ‚ p X , Q q , d “ 0 q is a ( q -)model for X . Spheres, Lie groups and their classifying spaces, homogeneous spaces G { K with r kG “ r kK , and K p π, n q ’s with n ě 2 are formal. Formality is preserved under (finite) direct products and wedges of spaces, as well as connected sums of manifolds. The 1-formality property of X depends only on π 1 p X q . (Macinic 2010) If X is a q -formal CW-complex of dimension at most q ` 1, then X is formal. A Koszul algebra is a graded k -algebra such that Tor A s p k , k q t “ 0 for all s ‰ t . (Papadima–Yuzvinsky 1999) Suppose H ‚ p X , k q is a Koszul algebra. Then X is formal if and only if X is 1-formal. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 5 / 24
A LGEBRAIC MODELS AND FORMALITY G EOMETRY AND FORMALITY G EOMETRY AND FORMALITY (Stasheff 1983) Let X be a k -connected CW-complex of dimension n . If n ď 3 k ` 1, then X is formal. (Miller 1979) If M is a closed, k -connected manifold of dimension n ď 4 k ` 2, then M is formal. In particular, all simply-connected, closed manifolds of dimension at most 6 are formal. (Fernández–Muñoz 2004) There exist closed, simply-connected, non-formal manifolds of dimension 7. (Deligne–Griffiths–Morgan–Sullivan 1975) All compact Kähler manifolds are formal. (Papadima–S. 2015) If M is a compact Sasakian manifold of dimension 2 n ` 1, then M is p 2 n ´ 1 q -formal. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 6 / 24
A LGEBRAIC MODELS AND FORMALITY P URITY IMPLIES FORMALITY P URITY IMPLIES FORMALITY (Morgan 1978) Let X be a smooth, quasi-projective variety. If W 1 H 1 p X , C q “ 0, then X is 1-formal. (Dupont 2016) More generally, suppose either H k p X q is pure of weight k , for all k ď q ` 1, or H k p X q is pure of weight 2 k , for all k ď q . Then X is q -formal. In particular, complements of hypersurfaces in CP n are 1-formal. Thus, complements of plane algebraic curves are formal. Complements of linear and toric arrangements are formal, but complements of elliptic arrangements may be non-1-formal. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 7 / 24
C OHOMOLOGY JUMP LOCI R ESONANCE VARIETIES OF A CDGA R ESONANCE VARIETIES OF A CDGA Assume the cdga p A , d q is connected, i.e., A 0 “ k , and of finite-type, i.e., dim A i ă 8 for all i ě 0. For each a P Z 1 p A q – H 1 p A q , we have a cochain complex, δ 0 δ 1 δ 2 a � A 1 a � A 2 a � ¨ ¨ ¨ , p A ‚ , δ a q : A 0 with differentials δ i a p u q “ a ¨ u ` d u , for all u P A i . The resonance varieties of p A , d q are the sets R i p A q “ t a P H 1 p A q | H i p A ‚ , δ a q ‰ 0 u . An element a P H 1 p A q belongs to R i p A q if and only if rank δ i ` 1 ` rank δ i a ă b i p A q . a If d “ 0, then the resonance varieties of A are homogeneous. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 8 / 24
C OHOMOLOGY JUMP LOCI C OHOMOLOGY JUMP LOCI OF SPACES C OHOMOLOGY JUMP LOCI OF SPACES The resonance varieties of a connected, finite-type CW-complex X are the subsets R i p X q : “ R i p H ‚ p X , C q , d “ 0 q of H 1 p X , C q . The variety R 1 p X q depends only on the group G “ π 1 p X q ; in fact, only on the second nilpotent quotient G { γ 3 p G q . The characteristic varieties of X are the Zariski closed sets of the character group of G given by V i p X q “ t ρ P Hom p G , C ˚ q | H i p X , C ρ q ‰ 0 u . The variety V 1 p X q depends only on the group G “ π 1 p X q ; in fact, only on the second derived quotient G { G 2 . Given any subvariety W Ă p C ˚ q n , there is a finitely presented group G such that G ab “ Z n and V 1 p G q “ W . A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 9 / 24
C OHOMOLOGY JUMP LOCI T HE T ANGENT C ONE THEOREM T HE T ANGENT C ONE THEOREM (Libgober 2002, Dimca–Papadima–S. 2009) τ 1 p V i p X qq Ď TC 1 p V i p X qq Ď R i p X q . Here, if W Ă p C ˚ q n is an algebraic subset, then τ 1 p W q : “ t z P C n | exp p λ z q P W , for all λ P C u is a finite union of rationally defined linear subspaces of C n . (DPS 2009/DP 2014) If X is q -formal, then, for all i ď q , τ 1 p V i p X qq “ TC 1 p W i p X qq “ R i p X q . This theorem yields a very efficient formality test. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 10 / 24
C OHOMOLOGY JUMP LOCI T HE T ANGENT C ONE THEOREM E XAMPLE Let G “ x x 1 , x 2 | r x 1 , r x 1 , x 2 ssy . Then V 1 p π q “ t t 1 “ 1 u , and so TC 1 p V 1 p π qq “ t x 1 “ 0 u . But R 1 p π q “ C 2 , and so π is not 1-formal. E XAMPLE Let G “ x x 1 , . . . , x 4 | r x 1 , x 2 s , r x 1 , x 4 sr x ´ 2 2 , x 3 s , r x ´ 1 1 , x 3 sr x 2 , x 4 sy . Then R 1 p π q “ t z P C 4 | z 2 1 ´ 2 z 2 2 “ 0 u : a quadric which splits into two linear subspaces over R , but is irreducible over Q . Thus, π is not 1-formal. E XAMPLE Let Conf n p E q be the configuration space of n labeled points of an elliptic curve. Then " ˇ * ř n i “ 1 x i “ ř n ˇ i “ 1 y i “ 0 , p x , y q P C n ˆ C n ˇ R 1 p Conf n p E qq “ . ˇ x i y j ´ x j y i “ 0 , for 1 ď i ă j ă n For n ě 3, this is an irreducible, non-linear variety (a rational normal scroll). Hence, Conf n p E q is not 1-formal. A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 11 / 24
F ILTERED AND GRADED L IE ALGEBRAS A SSOCIATED GRADED L IE ALGEBRAS A SSOCIATED GRADED L IE ALGEBRAS The lower central series of a group G is defined inductively by γ 1 G “ G and γ k ` 1 G “ r γ k G , G s . This forms a filtration of G by characteristic subgroups. The LCS quotients, γ k G { γ k ` 1 G , are abelian groups. The group commutator induces a graded Lie algebra structure on à gr p G , k q “ k ě 1 p γ k G { γ k ` 1 G q b Z k . Assume G is finitely generated. Then gr p G , k q is also finitely generated (in degree 1) by gr 1 p G , k q “ H 1 p G , k q . For instance, if F n is the free group of rank n , then gr p F n ; k q is the free graded Lie algebra Lie p k n q . A LEX S UCIU (N ORTHEASTERN ) F ORMALITY NOTIONS B ANFF , J UNE 2017 12 / 24
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