Fukaya categories of symmetric products and bordered Heegaard-Floer - PowerPoint PPT Presentation

Fukaya categories of symmetric products and bordered Heegaard-Floer homology Denis AUROUX Massachusetts Institute of Technology (work in progress)

Ozsv´ ath-Szab´ o invariants as a TQFT? ⇒ Φ( X ) ∈ Z • closed 4-manifold X = ( � HF , HF + , HF − ) • closed 3-manifold Y = ⇒ HF ( Y ) abelian group • cobordism ∂X = Y 2 − Y 1 = ⇒ Φ( X ) : HF ( Y 1 ) → HF ( Y 2 ) ————————— • surface Σ = ⇒ category C (Σ)? • 3-manifold with boundary ∂Y = Σ = ⇒ object C ( Y ) ∈ C (Σ)? (e.g.: handlebody) • cobordism ∂Y = Σ 2 − Σ 1 = ⇒ functor C ( Y ) : C (Σ 1 ) → C (Σ 2 )? • pairing: Y = Y 1 ∪ Σ Y 2 = ⇒ HF ( Y ) ≃ hom C (Σ) ( C ( Y 1 ) , C ( Y 2 ))? see also: Perutz, Lekili 1

Bordered Heegaard-Floer homology (R. Lipshitz–P. Ozsv´ ath–D. Thurston) • F (marked, parameterized) surface = ⇒ A ( F ) differential algebra ⇒ � ∼ • Y 3-manifold with ∂Y → F = CFA ( Y ) right A ∞ -module over A ( F ) • Y ′ 3-manifold with ∂Y ′ ∼ ⇒ � CFD ( Y ′ ) left dg-module over A ( F ) → − F = • cobordisms = ⇒ bimodules over A ( F ) CF ( Y ∪ F Y ′ ) ≃ � � • Pairing theorem: � CFD ( Y ′ ) CFA ( Y ) ⊗ A ( F ) Lipshitz-Ozsv´ ath-Thurston define A ( F ) combinatorially, to encode behavior of holomorphic strips upon neck-stretching (SFT). Goal: Symplectic interpretation of A ( F ) and � CFA ( Y ) in terms of Fukaya categories of Sym k ( F ) . 2

The algebra A ( F, k ) (Lipshitz-Ozsv´ ath-Thurston) Describe F (genus g ) by a pointed matched circle: 4 g points 1 , . . . , 4 g carrying labels 1 , . . . , 2 g, 1 , . . . , 2 g A ( F, k ) (1 ≤ k ≤ 2 g ) generated by k -tuples of either � i � • upwards stands (“Reeb chords”) connecting pairs of points ( i < j ) j � i � • pairs of horizontal dotted lines such that the k source labels (resp. target labels) in { 1 , . . . , 2 g } are all distinct. View A ( F, k ) as a finite (differential) category with objects S k = k -element subsets of { 1 , . . . , 2 g } . (4) Example ( g = 2, k = 2): 8 8 r r (3) 7 7 r r (2) 6 6 r r (1) [ 5 2 5 5 morphism { 1 , 2 } → { 2 , 4 } . 8 ] = r r (4) 4 4 r r (3) 3 3 r r (2) 2 2 r r (1) 1 1 r r 3

The algebra A ( F, k ) (continued) Differential: sum all ways of smoothing one crossing (double-crossing ∼ 0). (4) (4) 8 8 8 8 8 8 8 8 8 8 8 8 r r r r r r r r r r r r (3) (3) 7 7 7 7 7 7 7 7 7 7 7 7 r r r r r r r r r r r r (2) (2) 6 6 6 6 6 6 6 6 6 6 6 6 r r r r r r r r r r r r ∂ ∂ �→ �→ (1) (1) 5 5 5 5 5 5 5 5 5 5 5 5 r r r r r r r r + r r + r r (4) (4) 4 4 4 4 4 4 4 4 4 4 4 4 r r r r r r r r r r r r (3) (3) 3 3 3 3 3 3 3 3 3 3 3 3 r r r r r r r r r r r r (2) (2) 2 2 2 2 2 2 2 2 2 2 2 2 r r r r r r r r r r r r (1) (1) 1 1 1 1 1 1 1 1 1 1 1 1 r r r r r r r r r r r r [ 5 2 [ 5 6 [ 1 2 3 [ 1 2 3 [ 1 2 3 8 ] 6 8 ] 7 5 4 ] 7 4 5 ] 5 7 4 ] 0 Product: concatenation (double-crossing ∼ 0) (4) (4) 8 8 8 8 8 8 8 8 8 8 r r r r r r r r r r (3) (3) 7 7 7 7 7 7 7 7 7 7 r r r r r r r r r r (2) (2) 6 6 6 6 6 6 6 6 6 6 r r r r r r r r r r �→ �→ 0 (1) (1) 5 5 5 5 5 5 5 5 5 5 r r r r r r r r r r (4) (4) 4 4 4 4 4 4 4 4 4 4 r r r r r r r r r r (3) (3) 3 3 3 3 3 3 3 3 3 3 r r r r r r r r r r (2) (2) 2 2 2 2 2 2 2 2 2 2 r r r r r r r r r r (1) (1) 1 1 1 1 1 1 1 1 1 1 r r r r r r r r r r [ 2 5 [ 5 2 [ 2 5 [ 2 1 [ 5 2 5 6 ] 8 ] 8 6 ] 6 ] 8 ] 4

A ( F, k ) vs. the Fukaya category of Sym k ( F ) z α 2 g α 1 Definition. For s ∈ S k , let D s = � α i ⊂ Sym k ( F ) . i ∈ s Let F ′ = relative Fukaya category of (Sym k ( F ) , { z } × Sym k − 1 ( F )). ( “partially wrapped” ) � Theorem 1. A ( F, k ) ≃ hom F ′ ( D s , D s ′ ) . s,s ′ ∈S k 5

Fukaya categories L, L ′ ⊂ ( M, ω ) compact exact Lagr. ⇒ hom( L, L ′ ) = CF ( L, L ′ ) = Z | L ∩ L ′ | 2 • Differential ∂ : hom( L 0 , L 1 ) → hom( L 0 , L 1 ) L 1 � ∂ ( p ) , q � counts pseudo-holomorphic strips q p L 0 • Product m 2 : hom( L 0 , L 1 ) ⊗ hom( L 1 , L 2 ) → hom( L 0 , L 2 ) q L 2 L 1 � m 2 ( p, q ) , r � counts pseudo-holomorphic triangles r p L 0 • Higher products m k ( A ∞ category) Partially wrapped case: (in progress, cf. also Abouzaid, Seidel) • ∂M contact, N ⊂ ∂M , ρ : ∂M → [0 , 1], ρ − 1 (0) = N • H ρ Hamiltonian on ˆ M = M ∪ [1 , ∞ ) × ∂M , s.t. H ρ ( r, y ) = ρ ( y ) r near ∞ H ρ “wraps” along Reeb flow of contact hypersurface { r = ρ − 1 } ≃ ∂M \ N , slowing down as one approaches N ⇒ perturb Floer homology by long-time flow of H ρ : for ∂L, ∂L ′ ⊂ ∂M \ N , hom F ′ ( L, L ′ ) = w → + ∞ CF ( φ wH ρ ( L ) , L ′ ) . lim 6

Partial wrapping in Sym k ( F ) Partial wrapping of D s = � ≃ � Ham α i rel. { z } × Sym k − 1 ( F ) gives D − α − ˜ i . s i ∈ s i ∈ s z z α 2 g α 1 α − α − α + α + ˜ 2 g · · · ˜ ˜ 1 · · · ˜ 1 2 g 7

s ≃ � α ± Floer theory for D ± i ∈ s ˜ i α − α − α − α − ˜ 2 g · · · ˜ ˜ 2 g · · · ˜ α 2 g 1 1 α 1 z α 2 g α 1 α + ˜ q 2 g 2 g α + ˜ q 1 1 α + ˜ α − α − α + α + q 2 g 2 g ˜ 2 g · · · ˜ ˜ 1 · · · ˜ 1 2 g α + ˜ q 1 1 Proof of Theorem 1: s ′ ) ∼ s , D + • CF ( D − = hom A ( F,k ) ( s, s ′ ) (gen. by k -tuples of intersections) • ∂ counts empty rectangles (“nice diagram”) • product m 2 counts unions of triangles (head-to-tail overlap only) • m k ≥ 3 ≡ 0 8

Generating the relative Fukaya category � 2 g � “Theorem” 2. The relative Fukaya category F ′ is generated by the k objects D s , s ⊆ { 1 , . . . , 2 g } , | s | = k . Hence, F ′ -mod ≃ A ( F, k ) -mod. Key: D s are “thimbles” for a Lefschetz fibration f k : Sym k ( F ) → C . 2:1 → C (with 2 g + 1 branch points). • Start with π : F − � 2 g +1 � Then f k : { z 1 , . . . , z k } �→ � π ( z i ) has nondegenerate critical points k = tuples of distinct critical points of π . • The thimbles (stable manifolds for ∇ Re ( f k )) are products of k arcs on F ( α 1 , . . . , α 2 g + one other = thimbles of π ); they generate F ′ [Seidel]. � 2 g � • Can reduce to sub-fibration f − 1 k ( U ), U ⊂ C , with thimbles = { D s } s ∈S k . k 9

The A ∞ -module � CFA ( Y ) β 1 β 2 β 3 z α a 2 g y x, y x, y x α a 1 α c 1 Σ, genus ¯ g ≥ g α a 1 , . . . , α a 2 g ; α c 1 , . . . , α c ¯ g − g β 1 , . . . , β ¯ g e.g., x · [ 6 4 7 ] = y . 10

The A ∞ -module � CFA ( Y ) β 1 β 2 β 3 z α a 2 g y x, y x, y x α a 1 α c 1 Σ, genus ¯ g ≥ g α a 1 , . . . , α a 2 g ; α c 1 , . . . , α c ¯ g − g β 1 , . . . , β ¯ g CFA ( Y ) ≃ � � CF ( T β , T c “Theorem” 3. α × D s ) (right A ( F, g ) -module). s ∈S g Note: A ( F, g ) and F (Sym g F, z × Sym g − 1 F ) embed into F (Sym ¯ g Σ , z × Sym ¯ g − 1 Σ) via T c α 11

The pairing theorem CF ( Y ∪ F Y ′ ) ≃ hom A ( F,g ) − mod ( � CFA ( − Y ′ ) , � “Theorem” 4. � CFA ( Y )) . (Equivalent to Lipshitz-Ozsv´ ath-Thurston’s pairing result) Main ingredients: • Extended Fukaya categories (“quilts”) [Wehrheim-Woodward]: α as generalized Lagrangian in (Sym g ( F ) , z × Sym g − 1 ( F )) view T β ◦ T c • Yoneda embedding (+ Theorem 2): hom A− mod ( T β ′ ◦ T c α ′ , T β ◦ T c α ) ≃ CF ( T β ◦ T c α , T β ′ ◦ T c α ′ ) • “ CF ( T β × T c α ′ , T β ′ × T c α ) ≃ CF ( T β ◦ T c α , T β ′ ◦ T c α ′ )” [Lekili-Perutz] 12