T HE WORK OF S TEFAN P APADIMA IN TOPOLOGY AND GEOMETRY Alexandru - PowerPoint PPT Presentation

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS T HE WORK OF ¸ S TEFAN P APADIMA IN TOPOLOGY AND GEOMETRY Alexandru Suciu Northeastern University Topology and Geometry: A conference in memory of Stefan Papadima ¸ IMAR, Bucharest May 28, 2018

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS I NTRODUCTION § The work of ¸ Stefan Papadima spans some four decades (1977–2017). § His research covered many areas of Algebraic, Geometric, and Differential Topology; Algebraic and Differential Geometry; Several Complex Variables; Group Theory; Lie Algebras; and Combinatorics. Bucharest 1980 § He published over 70 articles, many in top journals, with half a dozen papers still coming out. § The two of us collaborated on 28 papers, starting in late 1999 during a 6-week Research in Pairs at Oberwolfach, with the last one completed in November 2017.

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS Here are some of the themes from Papadima’s work: § Rational Homotopy Theory § Rational homotopy of Thom spaces § Formality of spaces and maps § Rational classification of differentiable manifolds § Rigidity properties of homogeneous spaces § Isometry-invariant geodesics § Closed manifolds and Artinian complete intersections § Rational K p π, 1 q spaces and Koszul Boston 2006 algebras § Finite algebraic models and actions of compact Lie groups

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS § Lie Algebras § Malcev Lie algebras § Holonomy Lie algebras § Chen Lie algebras § Homotopy Lie algebras and the Rescaling Formula § Infinitesimal finiteness obstructions § Discrete Groups § Braids and Campbell-Hausdorff invariants § Finite-type invariants for braid groups § Right-angled Artin groups § Bestvina–Brady groups § McCool groups § Finiteness properties for Torelli groups § Johnson filtration of automorphism Trieste 2006 groups

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS Venice 2007 § Hyperplane Arrangements § Hypersolvable arrangements § Decomposable arrangements § Homotopy theory of complements of arrangements § Minimality of arrangement complements § Milnor fibrations of arrangements

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS Nice 2009 § Cohomology Jump Loci and Representation Varieties § Germs of cohomology jump loci § The Tangent Cone Formula § Jump loci for quasi-projective manifolds § Vanishing resonance and representations of Lie algebras § Representation varieties and deformation theory § Higher rank cohomology jump loci § Naturality properties of embedded jump loci

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS A SSOCIATED GRADED L IE ALGEBRAS § Let G be a group. The lower central series of G is defined inductively by γ 1 p G q “ G , γ 2 p G q “ G 1 “ r G , G s , and γ k ` 1 p G q “ r γ k p G q , G s . § Then γ k p G q Ÿ G , and gr k p G q : “ γ k p G q{ γ k ` 1 p G q is abelian. Set à gr p G q “ gr k p G q . 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. § If G is finitely generated, then gr p G q is also finitely generated, by gr 1 p G q “ G ab . We let φ k p G q “ rank gr k p G q . § 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 s

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS M ALCEV L IE ALGEBRAS § The group-algebra Q G has a natural Hopf algebra structure, with comultiplication ∆ p g q “ g b g and counit ε : Q G Ñ Q . § (Quillen 1968) Let I “ ker ε . The I -adic completion Ý k Q G { I k is a filtered, complete Hopf algebra. y Q G “ lim Ð § An element x P x k G is called primitive if p ∆ x “ x p b 1 ` 1 p b x . The set of all such elements, m p G q “ Prim p y Q G q , with bracket r x , y s “ xy ´ yx , is a complete, filtered Lie algebra, called the Malcev Lie algebra of G . § Moreover, if we set gr Q p G q “ gr p G q b Q , then gr p m p G qq – gr Q p G q . § (Sullivan 1977) A finitely genetared group G is 1-formal if and only if m p G q is a quadratic Lie algebra.

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS H OLONOMY L IE ALGEBRAS § Let G be a finitely generated group, with G ab torsion-free. § Set A i “ H i p G , Z q and A i “ p A i q ˚ “ Hom p A i , Z q . § The cup-product map A 1 b A 1 Ñ A 2 factors through a linear map µ : A 1 ^ A 1 Ñ A 2 . § Dualizing, and identifying p A 1 ^ A 1 q ˚ – A 1 ^ A 1 , we obtain a linear map, µ ˚ : A 2 Ñ A 1 ^ A 1 “ Lie 2 p A 1 q . D EFINITION ( C HEN 1973, M ARKL –P APADIMA 1992) The holonomy Lie algebra of G is h p G q “ Lie p A 1 q{x im µ ˚ y . § h p G q inherits a natural grading from Lie p A 1 q . § h p G q is a quadratic Lie algebra. § There is a canonical surjection h p G q ։ gr p G q , which is an isomorphism precisely when gr p G q is quadratic.

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS C HEN L IE ALGEBRAS § The Chen Lie algebra of a group G is gr p G { G 2 q , the associated graded Lie algebra of its maximal metabelian quotient. § Assuming G is finitely generated, write θ k p G q “ rank gr k p G { G 2 q for the Chen ranks. ` n ` k ´ 2 ˘ § (Chen 1951) θ k p F n q “ p k ´ 1 q , for all k ě 2. k § The projection G ։ G { G 2 induces gr p G q ։ gr p G { G 2 q , and so φ k p G q ě θ k p G q , with equality for k ď 3. § The map h p G q ։ gr p G q induces h p G q{ h p G q 2 ։ gr p G { G 2 q . T HEOREM ( P APADIMA –S. 2004) If G is 1 -formal, then h Q p G q{ h Q p G q 2 » Ñ gr Q p G { G 2 q . Ý Further improvements can be found in [S.–He Wang, 2017].

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS L IE ALGEBRAS OF A RAAG Let G “ G Γ “ x v P V p Γ q | vw “ wv if t v , w u P E p Γ qy be the right-angled Artin group associated to a finite simple graph Γ . T HEOREM ( D UCHAMP –K ROB 1992, P APADIMA –S. 2006) § gr p G q – h p G q . § The graded pieces are torsion-free, with ranks given by ś 8 k “ 1 p 1 ´ t k q φ k “ P Γ p´ t q , where P Γ p t q “ ř k ě 0 f k p Γ q t k is the clique polynomial of Γ , with f k p Γ q “ # t k-cliques of Γ u . T HEOREM ( PS 2006) § gr p G { G 2 q – h p G q{ h p G q 2 . § The graded pieces are torsion-free, with ranks given by ` ˘ ř 8 , where Q Γ p t q “ ř k “ 2 θ k t k “ Q Γ j ě 2 c j p Γ q t j is the t {p 1 ´ t q “cut polynomial" of Γ , with c j p Γ q “ ř W Ă V : | W |“ j ˜ b 0 p Γ W q .

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS T HE R ESCALING F ORMULA Let X be a connected space, and let Y be a simply-connected space (all spaces » to finite-type CW-complexes) D EFINITION ( P APADIMA –S. 2004) We say Y is a k-rescaling of X (over a ring R ) if: H ˚ p Y , R q – H ˚ p X , R qr k s as graded rings that is, H i p Y , R q – H j p X , R q if i “ p 2 k ` 1 q j and vanishes otherwise, and all isomorphisms compatible with cup products. Examples of rescalings (over R “ Z ) § X “ S 1 , Y “ S 2 k ` 1 1 S 1 ˆ S 1 , Y “ # g 1 S 2 k ` 1 ˆ S 2 k ` 1 § X “ # g § X “ C ℓ z Ť n i “ 1 H i , Y “ C p k ` 1 q ℓ z Ť n i “ 1 H ˆp k ` 1 q , where i A “ t H 1 , . . . , H n u is a hyperplane arrangement in C ℓ and A k ` 1 : “ t H ˆp k ` 1 q , . . . , H ˆp k ` 1 q u (the redundant subspace arr.) n 1

I NTRODUCTION G ROUPS AND L IE ALGEBRAS C OHOMOLOGY JUMP LOCI F INITENESS OBSTRUCTIONS A RRANGEMENTS § For a graded Lie algebra L , its k -rescaling is the graded Lie algebra L r k s with L r k s 2 kq “ L q and L r k s p “ 0 otherwise, and with Lie bracket rescaled accordingly. § The homotopy Lie algebra of a simply-connected space Y is the graded Lie algebra π ˚ p Ω Y q b Q : “ À r ě 1 π r p Ω Y q b Q , with Lie bracket coming from the Whitehead product. T HEOREM ( PS 2004) Let Y be a k-rescaling of X, and suppose H ˚ p X , Q q is a Koszul algebra. Then: § π ˚ p Ω Y q b Q – gr ˚ p π 1 X q b Q r k s . § Set Φ r : “ rank π r p Ω Y q “ rank π r ` 1 p Y q . Then Φ r “ 0 if 2 k ∤ r, and ź ` 1 ´ t p 2 k ` 1 q i ˘ Φ 2 ki “ Poin X p´ t k q . i ě 1 Consequently, Poin Ω Y p t q “ Poin X p´ t 2 k q ´ 1 .