Fukaya categories and bordered Heegaard-Floer homology Denis Auroux - PowerPoint PPT Presentation

Fukaya categories and bordered Heegaard-Floer homology Denis Auroux UC Berkeley / MIT International Congress of Mathematicians 2010 – Hyderabad arXiv:1001.4323 (to appear in J. G¨ okova Geom. Topol.) arXiv:1003.2962 (Proc. ICM 2010) 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 / MIT) Fukaya categories and Heegaard-Floer ICM 2010 1 / 12

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 / MIT) Fukaya categories and Heegaard-Floer ICM 2010 2 / 12

Heegaard-Floer TQFT Extend Heegaard-Floer to surfaces and 3-manifolds with boundary? 2 answers: Lipshitz-Ozsv´ ath-Thurston ’08 (explicit, computable) vs. Lekili-Perutz ’10 (geometric, can be extended to HF ± ) Y 3 closed � � HF ( Y ) abelian group W 4 cobordism ( ∂ W = Y 2 − Y 1 ) � � F W : � HF ( Y 1 ) → � HF ( Y 2 ) Σ surface (punctured, decorated) � category C (Σ) (Γ Σ acts faithfully) (modules over) finite dg-algebra A (Σ) (extended, balanced) Fukaya category F # (Sym g (Σ)) Y 3 with boundary ∂ Y = Σ � object C ( Y ) ∈ C (Σ)? � CFA ( Y ) right A ∞ A (Σ)-module (also: � CFD ( Y ) left dg-module) T Y (generalized) Lagrangian submanifold of Sym g (Σ) cobordism ∂ Y = Σ 2 − Σ 1 � functor C (Σ 1 ) → C (Σ 2 ) from bimodule � CFDA ( Y ), (generalized) Lagr. correspondence T Y HF ( Y 1 ∪ Σ Y 2 ) = hom mod- A ( � � Y 2 ) , � CFA ( − CFA ( Y 1 )) = HF ( T Y 1 , T - Y 2 ) Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 3 / 12

Goal: relate these two approaches Plan Background: Floer homology, Fukaya categories, correspondences The Lekili-Perutz approach: correspondences from cobordisms The Lipshitz-Ozsv´ ath-Thurston strands algebra The partially wrapped Fukaya category of Sym k (Σ) Modules and bimodules from bordered 3-manifolds Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 4 / 12

Floer homology, Fukaya categories and correspondences Σ Riemann surface, M = Sym k (Σ) monotone symplectic manifold � Fukaya category F ( M ): objects = Lagrangian submanifolds ∗ (closed) hom( L , L ′ ) = CF ( L , L ′ ) = � (monotone, balanced) x ∈ L ∩ L ′ Z 2 x L ′ differential ∂ : CF ( L , L ′ ) → CF ( L , L ′ ) y x coeff. of y in ∂ x counts holom. strips L composition CF ( L , L ′ ) ⊗ CF ( L ′ , L ′′ ) → CF ( L , L ′′ ) y L ′′ L ′ coeff. of z in x · y counts holom. triangles z x more ( A ∞ -category) L (for product Lagrangians, holom. curves in Sym k (Σ) can be seen on Σ) L Lagrangian correspondences M 1 − → M 2 = Lagrangian submanifolds L ⊂ ( M 1 × M 2 , – ω 1 ⊕ ω 2 ) generalize symplectomorphisms. “generalized Lagrangians” = formal images of Lagrangians under sequences of correspondences; Floer theory extends well. � extended Fukaya cat. F # ( M ) (Ma’u-Wehrheim-Woodward). Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 5 / 12

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 / MIT) Fukaya categories and Heegaard-Floer ICM 2010 6 / 12

The Lipshitz-Ozsv´ ath-Thurston strands algebra A (Σ , k ) 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 , k ) is generated (over Z 2 ) by k -tuples of { upward strands, pairs of horizontal dotted lines } s.t. the k source labels (resp. target labels) in { 1 , . . . , 2 g } are all distinct. Example ( g = k = 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 / MIT) Fukaya categories and Heegaard-Floer ICM 2010 7 / 12

The extended Fukaya category vs. A (Σ , k ) Theorem F # (Sym k (Σ)) embeds fully faithfully into mod- A (Σ , k ) (A ∞ -modules) Main tool: partially wrapped Fukaya cat. F # (Sym k (Σ) , z ) ( z ∈ ∂ Σ) Enlarge F # : allow noncompact objects = products of k disjoint properly embedded arcs; Floer theory perturbed by Hamiltonian flow. Roughly: hom( L 0 , L 1 ) := CF (˜ L 0 , ˜ L 1 ), isotoping arcs so that end points of ˜ L 0 lie above those of ˜ L 1 in ∂ Σ \ { z } (without crossing z ) Similarly, product is defined by perturbing so that ˜ L 0 > ˜ L 1 > ˜ L 2 . (after Abouzaid-Seidel) Let D s = � α i ( s ⊆ { 1 ... 2 g } , | s | = k ). Then: z α 2 g i ∈ s 1. � α 1 hom( D s , D t ) ≃ A (Σ , k ) s , t 2. the objects D s generate F # (Sym k (Σ) , z ) Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 8 / 12

� hom( D s , D t ) ≃ A (Σ , k ) � � s = � By def. of F # (Sym k (Σ) , z ), hom( D s , D t ) = CF (˜ s , ˜ D − ˜ α ± D + D ± t ) ˜ i i ∈ s z α + α + α − α − ˜ 2 g · · · ˜ ˜ 1 · · · ˜ 1 2 g � α + α − j Dictionary: points of ˜ i ∩ ˜ j ← → strands q generators = k -tuples i q ) q (intersections on central axis ← → q q q y l x l l q q Differential: y appears in ∂ x iff ← → x = and y = k k j i q q j j q q x y i i k q q Similarly for product (triple diagram); all diagrams are “nice” More generally: Z ⊂ ∂ Σ finite, α i ⊂ Σ disjoint arcs s.t. each component of Σ \ � α i contains ≥ 1 point of Z . Let D s = � i ∈ s α i ∈ F # (Sym k Σ , Z ). Then � hom( D s , D t ) is a combinatorially explicit, LOT-type, dg-algebra. Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 9 / 12

{ D s = � i ∈ s α i } s ⊆{ 1 ... 2 g } generate F # (Sym k (Σ) , z ) 2:1 → C induces a Lefschetz fibration f k : Sym k (Σ) → C with π : Σ − � 2 g +1 � critical points. Its thimbles = products of α i (1 ≤ i ≤ 2 g + 1) k generate F ( f k ) ≃ F (Sym k Σ , { z , z ′ } ) (Seidel) z α 2 g +1 α 2 g α 1 z ′ � 2 g +1 � objects also generate F # (Sym k Σ , z ). These k Uses: acceleration functor F (Sym k Σ , { z , z ′ } ) → F (Sym k Σ , z ) (Abouzaid-Seidel) α i 1 × · · · × α 2 g +1 ≃ twisted complex built from { α i 1 × · · · × α j } 2 g j =1 Uses: arc slides are mapping cones More generally: Z ⊂ ∂ Σ finite, α i ⊂ Σ disjoint arcs s.t. each component of Σ \ � α i is a disc containing ≤ 1 point of Z . Then the products D s = � i ∈ s α i generate F # (Sym k Σ , Z ). Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 10 / 12

Yoneda embedding and A ∞ -modules Recall: Y 3 , ∂ Y = Σ ∪ D 2 ⇒ gen. Lagr. T Y ∈ F # (Sym g Σ) (Lekili-Perutz) Yoneda embedding: T Y �→ Y ( T Y ) = � s hom( T Y , D s ) right A ∞ -module over � s , t hom( D s , D t ) ≃ A (Σ , g ). In fact, Y ( T Y ) ≃ � CFA ( Y ) (bordered Heegaard-Floer module) Pairing theorem: if Y = Y 1 ∪ Y 2 , ∂ Y 1 = − ∂ Y 2 = Σ ∪ D 2 , then � CF ( Y ) ≃ hom F # ( T Y 1 , T − Y 2 ) ≃ hom mod- A ( Y ( T − Y 2 ) , Y ( T Y 1 )) . also: (using A ( − Σ , g ) ≃ A (Σ , g ) op ) � CF ( Y ) ≃ T Y 1 ◦ T Y 2 ≃ Y ( T Y 1 ) ⊗ A Y ( T Y 2 ). More generally, if ∂ Y = Σ + ∪ − Σ − (sutured manifold), the generalized corresp. T Y ∈ F # ( − Sym k − Σ − × Sym k + Σ + ) yields an A ∞ -bimodule Y ( T Y ) = � s , t hom( D − , s , T Y , D + , t ) ∈ A (Σ − , k − )-mod- A (Σ + , k + ) (cf. Ma’u-Wehrheim-Woodward). Y ( T Y ) ≃ � CFDA ( Y )? (same properties) Denis Auroux (UC Berkeley / MIT) Fukaya categories and Heegaard-Floer ICM 2010 11 / 12