The symplectic geometry of symmetric products and invariants of 3-manifolds with boundary Denis Auroux UC Berkeley AMS Invited Address – Joint Mathematics Meetings New Orleans, January 2011 builds on work of: R. Lipshitz, P. Ozsv´ ath, D. Thurston; T. Perutz, Y. Lekili M. Abouzaid, P. Seidel; S. Ma’u, K. Wehrheim, C. Woodward Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 1 / 13
Low-dimensional topology Goal: find invariants to distinguish smooth manifolds Dimensions 3 and 4 hardest (Poincar´ e conjecture, . . . ) Exotic smooth 4-manifolds (homeomorphic, not diffeomorphic) Smooth 3- and 4-manifold invariants (beyond algebraic topology) 80’s Donaldson invariants 90’s Seiberg-Witten invariants increasingly computable and versatile 00’s Ozsv´ ath-Szab´ o invariants ❄ These all associate numerical invariants to closed 4-manifolds, and (graded) abelian groups to closed 3-manifolds. But the story goes further! Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 2 / 13
Heegaard-Floer TQFT Ozsv´ ath-Szab´ o (2000) Y 3 closed � � HF ( Y ) abelian group (Heegaard-Floer homology) W 4 cobordism ( ∂ W = Y 2 − Y 1 ) � � F W : � HF ( Y 1 ) → � HF ( Y 2 ) (and more) Extend to surfaces and 3-manifolds with boundary? Σ surface � category C (Σ)? Y 3 with boundary ∂ Y = Σ � object C ( Y ) ∈ C (Σ)? cobordism ∂ Y = Σ 2 − Σ 1 � functor C (Σ 1 ) → C (Σ 2 )? Want: � HF ( Y 1 ∪ Σ Y 2 ) = hom C (Σ) ( C ( Y 1 ) , C ( Y 2 )) (pairing theorem) This can be done in 2 equivalent ways: bordered Heegaard-Floer homology (Lipshitz-Ozsv´ ath-Thurston 2008, more computable), or geometry of Lagrangian correspondences (Lekili-Perutz 2010, more conceptual). Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 3 / 13
Plan of the talk Heegaard-Floer homology Background: Floer homology, Fukaya categories, correspondences The Lekili-Perutz approach: correspondences from cobordisms The Fukaya category of the symmetric product The Lipshitz-Ozsv´ ath-Thurston strands algebra Modules and bimodules from bordered 3-manifolds Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 4 / 13
Heegaard-Floer homology Y 3 closed 3-manifold admits a Heegaard H β splitting into two handlebodies Y = H α ∪ ¯ Σ H β . This is encoded by a Heegaard diagram (¯ Σ , α 1 . . . α g , β 1 . . . β g ). ( g = genus (¯ Σ)) β 1 β g H α z ¯ Σ α 1 α g unordered g -tuples of points on punctured Σ Let T α = α 1 × · · · × α g , T β = β 1 × · · · × β g ⊂ Sym g (¯ Σ \ z ) Theorem ( Ozsv´ ath-Szab´ o, ∼ 2000) � HF ( Y ) := HF ( T β , T α ) is independent of chosen Heegaard diagram. (Floer homology: complex generated by T α ∩ T β = g -tuples of intersections between α and β curves, differential counts holomorphic curves). Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 5 / 13
Floer homology and Fukaya categories Σ Riemann surface � M = Sym g (Σ) symplectic manifold (monotone) Products of disjoint loops/arcs (e.g. T α = α 1 × · · · × α g ) are Lagrangian. Floer homology = Lagrangian intersection theory, corrected by holomorphic discs to ensure deformation invariance. Floer complex CF ( L , L ′ ) = � x ∈ L ∩ L ′ Z 2 x (assuming L , L ′ transverse) differential ∂ : CF ( L , L ′ ) → CF ( L , L ′ ) L ′ y x coeff. of y in ∂ x counts holomorphic strips L HF ( L , L ′ ) = Ker ∂/ Im ∂ . (For product Lagrangians T α , T β ⊂ Sym g (Σ), intersections = tuples of α i ∩ β σ ( i ) ; holom. curves in Sym g (Σ) can be seen on Σ. So � HF = HF ( T β , T α ) fairly easy) Fukaya category F ( M ): objects = Lagrangian submanifolds ∗ (closed) (monotone, balanced) hom( L , L ′ ) = CF ( L , L ′ ) with differential ∂ y L ′′ L ′ composition CF ( L , L ′ ) ⊗ CF ( L ′ , L ′′ ) → CF ( L , L ′′ ) z x coeff. of z in x · y counts holom. triangles L more ( A ∞ -category) Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 6 / 13
Lagrangian correspondences; the Lekili-Perutz TQFT L Lagrangian correspondences M 1 − → M 2 = Lagrangian submanifolds L ⊂ ( M 1 × M 2 , – ω 1 ⊕ ω 2 ). These generalize symplectomorphisms (but need not be single-valued); should map Lagrangians to Lagrangians. “Generalized Lagrangians” = formal images of Lagrangians under sequences of correspondences; Floer theory extends well. � extended Fukaya cat. F # ( M ) (Ma’u-Wehrheim-Woodward). L → M 2 induce functors F # ( M 1 ) → F # ( M 2 ). Correspondences M 1 − Heegaard-Floer TQFT Σ (punctured) surface � category C (Σ) = F # (Sym g (Σ)) Y 3 with boundary ∂ Y = Σ � object T Y : (generalized) Lagrangian submanifold of Sym g (Σ) (for a handlebody, T Y = product torus) cobordism ∂ Y = Σ 2 − Σ 1 � functor induced by (generalized) Lagr. correspondence T Y : Sym k 1 (Σ 1 ) − → Sym k 2 (Σ 2 ). Pairing theorem: � HF ( Y 1 ∪ Σ Y 2 ) ≃ HF ( T Y 1 , T − Y 2 ). Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 7 / 13
Lekili-Perutz: correspondences from cobordisms Σ − = Σ 0 Perutz: Elementary cobordism Y 12 : Σ 1 � Σ 2 = ⇒ Lagrangian correspondence Y 01 T 12 ⊂ Sym k (Σ 1 ) × Sym k +1 (Σ 2 ) ( k ≥ 0) Σ 1 (roughly: k points on Σ 1 �→ “same” k points on Σ 2 Y 12 plus one point anywhere on γ ) Lekili-Perutz: decompose Y 3 into sequence of γ Σ 2 elementary cobordisms Y i , i +1 , compose all T i , i +1 to get a generalized correspondence T Y . . . . Y : Sym k − (Σ − ) → Sym k + (Σ + ) ( ∂ Y =Σ + − Σ − ) T Σ + Theorem (Lekili-Perutz) T Y is independent of decomposition of Y into elementary cobordisms. View Y 3 ( sutured: ∂ Y =Σ + ∪ Σ − ) as cobordism of surfaces w. boundary For a handlebody (as cobordism D 2 � Σ g ), T Y ≃ product torus Y 3 closed, Y \ B 3 : D 2 � D 2 , then T Y ≃ � HF ( Y ) ∈ F # ( pt ) = Vect Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 8 / 13
Lekili-Perutz vs. bordered Heegaard-Floer The extended Fukaya category F # (Sym g (Σ)) and the generalized Lagrangians T Y (for Y 3 with ∂ Y = Σ) constructed by Lekili-Perutz are not very explicit at first glance... unlike Bordered Heegaard-Floer homology (Lipshitz-Ozsv´ ath-Thurston 2008) Σ (decorated) surface � (cat. of modules over) dg-algebra A (Σ , g ) Y 3 with ∂ Y = Σ � � CFA ( Y ) (right A ∞ ) module over A (Σ , g ) pairing: � HF ( Y 1 ∪ Σ Y 2 ) ≃ hom mod- A ( � Y 2 ) , � CFA ( − CFA ( Y 1 )) In fact, by considering specific product Lagrangians in Sym g (Σ) one gets: Theorem F # (Sym g (Σ)) embeds fully faithfully into mod- A (Σ , g ) Given Y 3 with ∂ Y = Σ , the embedding maps T Y to � CFA ( Y ) Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 9 / 13
The Lipshitz-Ozsv´ ath-Thurston strands algebra A (Σ , g ) Describe Σ by a pointed matched circle: segment with 4 g points carrying labels 1 , . . . , 2 g , 1 , . . . , 2 g (= how to build Σ = D 2 ∪ 2 g 1-handles) A (Σ , g ) is generated (over Z 2 ) by g -tuples of { upward strands, pairs of horizontal dotted lines } s.t. the g source labels (resp. target labels) in { 1 , . . . , 2 g } are all distinct. Example ( g = 2) 4 4 4 4 4 4 4 4 4 4 q q q q q q q q q q 3 3 3 3 3 3 3 3 3 3 q q q q q q q q q q 2 2 2 2 2 2 2 2 2 2 ∂ q q q q q q q q q q �→ �→ 1 1 1 1 1 1 1 1 1 1 q q q q q q q q q q 4 4 4 4 4 4 4 4 4 4 q q q q q q q q q q 3 3 3 3 3 3 3 3 3 3 q q q q q q q q q q 2 2 2 2 2 2 2 2 2 2 q q q q q q q q q q 1 1 1 1 1 1 1 1 1 1 q q q q q q q q q q { 1 , 2 } �→ { 2 , 4 } Differential: sum all ways of smoothing one crossing. Product: concatenation (end points must match). q as q q + q q Treat q and set q = 0. q q q q q q q q q Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 10 / 13
The extended Fukaya category vs. A (Σ , g ) Theorem F # (Sym g (Σ)) embeds fully faithfully into mod- A (Σ , g ) (A ∞ -modules) Main tool: partially wrapped Fukaya cat. F # (Sym g (Σ) , z ) ( z ∈ ∂ Σ) Enlarge F # : add noncompact objects = products of disjoint properly embedded arcs. Roughly, hom( L 0 , L 1 ) := CF (˜ L 0 , ˜ L 1 ), deforming all arcs so that end points of ˜ L 0 lie above those of ˜ L 1 (without crossing z ). Similarly, product is defined by perturbing so that ˜ L 0 > ˜ L 1 > ˜ L 2 . (after Abouzaid-Seidel) Let D s = � z α i ( s ⊆ { 1 ... 2 g } , | s | = g ). Then: α 2 g i ∈ s 1. � α 1 hom( D s , D t ) ≃ A (Σ , g ) s , t 2. the objects D s generate F # (Sym g (Σ) , z ) Denis Auroux (UC Berkeley) Symmetric products & 3-manifold invariants JMM, New Orleans, Jan. 2011 11 / 13
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