F INITENESS AND FORMALITY OBSTRUCTIONS Alex Suciu Northeastern University Workshop on Polyhedral Products in Homotopy Theory The Fields Institute, Toronto January 21, 2020 A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 1 / 27
R ESONANCE VARIETIES OF A CDGA R ESONANCE VARIETIES OF A CDGA Let A “ p A ‚ , d q be a commutative, differential graded algebra over a field k of characteristic 0. That is: 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 . We assume A is connected (i.e., A 0 “ k ¨ 1) and of finite-type (i.e., dim A i ă 8 for all i ). 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 p u q , for all u P A i . The resonance varieties of A are the affine varieties R i s p A q “ t a P H 1 p A q | dim k H i p A ‚ , δ a q ě s u . A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 2 / 27
R ESONANCE VARIETIES OF A CDGA Fix a k -basis t e 1 , . . . , e r u for A 1 , and let t x 1 , . . . , x r u be the dual basis for A 1 “ p A 1 q ˚ . Identify Sym p A 1 q with S “ k r x 1 , . . . , x r s , the coordinate ring of the affine space A 1 . Build a cochain complex of free S -modules, L p A q : “ p A ‚ b S , δ q : δ i δ i ` 1 � A i ` 2 b S � A i b S � A i ` 1 b S � ¨ ¨ ¨ , ¨ ¨ ¨ where δ i p u b f q “ ř r j “ 1 e j u b fx j ` d u b f . The specialization of p A b S , δ q at a P Z 1 p A q is p A , δ a q . Hence, R i s p A q is the zero-set of the ideal generated by all minors of size b i p A q ´ s ` 1 of the block-matrix δ i ` 1 ‘ δ i . A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 3 / 27
C HARACTERISTIC VARIETIES C HARACTERISTIC VARIETIES Let X be a connected, finite-type CW-complex. Then π “ π 1 p X , x 0 q is a finitely presented group, with π ab – H 1 p X , Z q . The ring R “ C r π ab s is the coordinate ring of the character group, Char p X q “ Hom p π, C ˚ q – p C ˚ q r ˆ Tors p π ab q , where r “ b 1 p X q . The characteristic varieties of X are the homology jump loci V i s p X q “ t ρ P Char p X q | dim C H i p X , C ρ q ě s u . These varieties are homotopy-type invariants of X , with V 1 s p X q depending only on π “ π 1 p X q . Set V 1 1 p π q : “ V 1 1 p K p π, 1 qq ; then V 1 1 p π q “ V 1 p π { π 2 q . E XAMPLE Let f P Z r t ˘ 1 1 , . . . , t ˘ 1 n s be a Laurent polynomial, f p 1 q “ 0. There is then a finitely presented group π with π ab “ Z n such that V 1 1 p π q “ V p f q . A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 4 / 27
T HE T ANGENT C ONE THEOREM T ANGENT CONES Let exp: H 1 p X , C q Ñ H 1 p X , C ˚ q be the coefficient homomorphism induced by C Ñ C ˚ , z ÞÑ e z . Let W “ V p I q , a Zariski closed subset of Char p G q “ H 1 p X , C ˚ q . The tangent cone at 1 to W is TC 1 p W q “ V p in p I qq . The exponential tangent cone at 1 to W : τ 1 p W q “ t z P H 1 p X , C q | exp p λ z q P W , @ λ P C u . Both tangent cones are homogeneous subvarieties of H 1 p X , C q ; are non-empty iff 1 P W ; depend only on the analytic germ of W at 1; commute with finite unions and arbitrary intersections. τ 1 p W q Ď TC 1 p W q , with “ if all irred components of W are subtori, but ‰ in general. (Dimca–Papadima–S. 2009) τ 1 p W q is a finite union of rationally defined subspaces. A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 5 / 27
A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES A CDGA map ϕ : A Ñ B is a quasi-isomorphism if ϕ ˚ : H . p A q Ñ H . p B q is an isomorphism. 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 CDGA s, A and B , are (q-) equivalent if there is a zig-zag of ( q -) quasi-isomorphisms connecting A to B . A is formal (or just q-formal ) if it is ( q -) equivalent to p H ‚ p A q , d “ 0 q . A CDGA is q-minimal if it is of the form p Ź V , d q , 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 p A q “ k admits a q-minimal model , M q p A q (i.e., a q -equivalence M q p A q Ñ A with M q p A q “ p Ź V , d q a q -minimal cdga), unique up to iso. A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 6 / 27
A LGEBRAIC MODELS FOR SPACES Given any (path-connected) space X , there is an associated Sullivan Q -cdga, A PL p X q , such that 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 -) equivalent to A PL p X q b Q k . If M is a smooth manifold, then Ω dR p M q 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 ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 7 / 27
A LGEBRAIC MODELS FOR SPACES T HE TANGENT CONE THEOREM Let X be a connected CW-complex with finite q -skeleton. Suppose X admits a q -finite q -model A . T HEOREM For all i ď q and all s: (DPS 2009, Dimca–Papadima 2014) V i s p X q p 1 q – R i s p A q p 0 q . (Budur–Wang 2017) All the irreducible components of V i s p X q passing through the origin of Char p X q are algebraic subtori . Consequently, τ 1 p V i s p X qq “ TC 1 p V i s p X qq “ R i s p A q . T HEOREM (P APADIMA –S. 2017) A f.g. group G admits a 1 -finite 1 -model if and only if the Malcev Lie algebra m p G q is the LCS completion of a finitely presented Lie algebra. A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 8 / 27
I NFINITESIMAL FINITENESS OBSTRUCTIONS I NFINITESIMAL FINITENESS OBSTRUCTIONS T HEOREM Let X be a connected CW-complex with finite q-skeleton. Suppose X admits a q-finite q-model A. Then, for all i ď q and all s, (Dimca–Papadima 2014) V i s p X q p 1 q – R i s p A q p 0 q . In particular, if X is q-formal, then V i s p X q p 1 q – R i s p X q p 0 q . (Macinic, Papadima, Popescu, S. 2017) TC 0 p R i s p A qq Ď R i s p X q . (Budur–Wang 2017) All the irreducible components of V i s p X q passing through the origin of H 1 p X , C ˚ q are algebraic subtori . E XAMPLE Let G be a f.p. group with G ab “ Z n and V 1 1 p G q “ t t P p C ˚ q n | ř n i “ 1 t i “ n u . Then G admits no 1-finite 1-model. A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 9 / 27
I NFINITESIMAL FINITENESS OBSTRUCTIONS T HEOREM (P APADIMA –S. 2017) Suppose X is p q ` 1 q finite, or X admits a q-finite q-model. Then b i p M q p X qq ă 8 , for all i ď q ` 1 . C OROLLARY Let G be a f.g. group. Assume that either G is finitely presented, or G has a 1 -finite 1 -model. Then b 2 p M 1 p G qq ă 8 . E XAMPLE Consider the free metabelian group G “ F n { F 2 n with n ě 2. We have V 1 p G q “ V 1 p F n q “ p C ˚ q n , and so G passes the Budur–Wang test. But b 2 p M 1 p G qq “ 8 , and so G admits no 1-finite 1-model (and is not finitely presented). A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 10 / 27
L IE ALGEBRAS ATTACHED TO GROUPS L OWER CENTRAL SERIES L OWER CENTRAL SERIES Let G be a group. The lower central series t γ k p G qu k ě 1 is defined inductively by γ 1 p G q “ G and γ k ` 1 p G q “ r G , γ k p G qs . Here, if H , K ă G , then r H , K s is the subgroup of G generated by tr a , b s : “ aba ´ 1 b ´ 1 | a P H , b P K u . If H , K Ÿ G , then r H , K s Ÿ G . The subgroups γ k p G q are, in fact, characteristic subgroups of G . Moreover r γ k p G q , γ ℓ p G qs Ď γ k ` ℓ p G q , @ k , ℓ ě 1. γ 2 p G q “ r G , G s is the derived subgroup, and so G { γ 2 p G q “ G ab . r γ k p G q , γ k p G qs Ÿ γ k ` 1 p G q , and thus the LCS quotients, gr k p G q : “ γ k p G q{ γ k ` 1 p G q are abelian. If G is finitely generated, then so are its LCS quotients. Set φ k p G q : “ rank gr k p G q . A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 11 / 27
L IE ALGEBRAS ATTACHED TO GROUPS A SSOCIATED GRADED L IE ALGEBRA A SSOCIATED GRADED L IE ALGEBRA Fix a coefficient ring k . Given a group G , we let à gr p G , k q “ gr k p G q b k . k ě 1 This is a graded Lie algebra, with Lie bracket r , s : gr k ˆ gr ℓ Ñ gr k ` ℓ induced by the group commutator. For k “ Z , we simply write gr p G q “ gr p G , Z q . The construction is functorial. Example: if F n is the free group of rank n , then gr p F n q is the free Lie algebra Lie p Z n q . ř k gr k p F n q is free abelian, of rank φ k p F n q “ 1 d . d | k µ p d q n k A LEX S UCIU (N ORTHEASTERN ) F INITENESS & FORMALITY OBSTRUCTIONS J ANUARY 21, 2020 12 / 27
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