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Denis Auroux-Fukaya categories of symmetric products and bordered Heegaard-Floer homology

a r X i v : 1 0 0 1 . 4 3 2 3 v 3 [ m a t h . G T ] 2 8 J u l 2 0 1 0 Journal of G¨okova Geometry Topology Volume x (20xx) 1 – 54 Fukaya categories of symmetric products and bordered Heegaard-Floer homology Denis Auroux Abstract. The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsv´ ath and Thurston’s bordered Heegaard-Floer homology [8] in terms of Fukaya categories of symmetric products and L
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    a  r   X   i  v  :   1   0   0   1 .   4   3   2   3  v   3   [  m  a   t   h .   G   T   ]   2   8   J  u   l   2   0   1   0 Journal of G¨okova Geometry TopologyVolume x (20xx) 1 – 54 Fukaya categories of symmetric products andbordered Heegaard-Floer homology Denis Auroux  Abstract. The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsv´ath and Thurston’s bordered Heegaard-Floer homology [8] in termsof Fukaya categories of symmetric products and Lagrangian correspondences. Morespecifically, we give a description of the algebra A ( F  ) which appears in the work of Lipshitz, Ozsv´ath and Thurston in terms of (partially wrapped) Floer homology forproduct Lagrangians in the symmetric product, and outline how bordered Heegaard-Floer homology itself can conjecturally be understood in this language. 1. Introduction Lipshitz, Ozsv´ath and Thurston’s bordered Heegaard-Floer homology  [8] extends the hatversion of Heegaard-Floer homology to an invariant for 3-manifolds with parametrizedboundary. Their construction associates to a (marked and parametrized) surface F  acertain algebra A ( F  ), and to a 3-manifold with boundary F  a pair of ( A ∞ -)modules over A ( F  ), which satisfy a TQFT-like gluing theorem. On the other hand, recent work of Lekili and Perutz [5] suggests another construction, whereby a 3-manifold with boundaryyields an object in (a variant of) the Fukaya category of the symmetric product of  F  . 1.1. Lagrangian correspondences and Heegaard-Floer homology Given a closed 3-manifold Y  , the Heegaard-Floer homology group  HF  ( Y  ) is classicallyconstructed by Ozsv´ath and Szab´o from a Heegaard decomposition by considering theLagrangianFloer homology of two product tori in the symmetric product of the puncturedHeegaard surface. Here is an alternative description of this invariant.Equip Y  with a Morse function (with only one minimum and one maximum, andwith distinct critical values). Then the complement Y  ′ of a ball in Y  (obtained bydeleting a neighborhood of a Morse trajectory from the maximum to the minimum) canbe decomposed into a succession of elementary cobordisms Y  ′ i ( i = 1 ,...,r ) betweenconnected Riemann surfaces with boundary Σ 0 , Σ 1 ,..., Σ r (where Σ 0 = Σ r = D 2 , andthe genus increases or decreases by 1 at each step). By a construction of Perutz [11], each Y  ′ i determines a Lagrangian correspondence L i ⊂ Sym g i − 1 (Σ i − 1 ) × Sym g i (Σ i ) between Key words and phrases. Bordered Heegaard-Floer homology, partially wrapped Fukaya category.This work was partially supported by NSF grants DMS-0600148 and DMS-0652630. 1  Denis Auroux symmetric products. The quilted Floer homology  of the sequence ( L 1 ,...,L r ), as definedby Wehrheim and Woodward [17, 18], is then isomorphic to  HF  ( Y  ). (This relies ontwo results from the work in progress of Lekili and Perutz [5]: the first one concerns theinvariance of this quilted Floer homology under exchanges of critical points, which allowsone to reduce to the case where the genus first increases from 0 to g then decreases backto 0; the second one states that the composition of the Lagrangian correspondences fromSym 0 ( D 2 ) to Sym g (Σ g ) is then Hamiltonian isotopic to the product torus considered byOzsv´ath and Szab´o.)Given a 3-manifold Y  with boundary ∂Y  ≃ F  ∪ S 1 D 2 (where F  is a connected genus g surface with one boundary component), we can similarly view Y  as a succession of elementary cobordisms (from D 2 to F  ), and hence associate to it a sequence of Lagrangiancorrespondences ( L 1 ,...,L r ). This defines an object T Y  of the extended Fukaya category  F  ♯ (Sym g ( F  )), as defined by Ma’u, Wehrheim and Woodward [10] (see [17, 18] for thecohomology level version).More generally, we can consider a cobordism between two connected surfaces F  1 and F  2 (each with one boundary component), i.e., a 3-manifold Y  12 with connected bound-ary, together with a decomposition ∂Y  12 ≃ − F  1 ∪ S 1 F  2 . The same construction as-sociates to such Y  a generalized Lagrangian correspondence (i.e., a sequence of corre-spondences) from Sym k 1 ( F  1 ) to Sym k 2 ( F  2 ), whenever k 2 − k 1 = g ( F  2 ) − g ( F  1 ); by Ma’u,Wehrheim and Woodward’s formalism, such a correspondence defines an A ∞ -functor from F  ♯ (Sym k 1 ( F  1 )) to F  ♯ (Sym k 2 ( F  2 )).To summarize, this suggests that we should associate: ã to a genus g surface F  (with one boundary), the collection of extended Fukayacategories of its symmetric products, F  ♯ (Sym k ( F  )) for 0 ≤ k ≤ 2 g ; ã to a 3-manifold Y  with boundary ∂Y  ≃ F  ∪ S 1 D 2 , an object of  F  ♯ (Sym g ( F  ))(namely, the generalized Lagrangian T Y  ); ã to a cobordism Y  12 with boundary ∂Y  12 ≃ − F  1 ∪ S 1 F  2 , a collection of  A ∞ -functorsfrom F  ♯ (Sym k 1 ( F  1 )) to F  ♯ (Sym k 2 ( F  2 )).These objects behave naturally under gluing: for example, if a closed 3-manifold de-composes as Y  = Y  1 ∪ F  ∪ D 2 Y  2 , where ∂Y  1 = F  ∪ D 2 = − ∂Y  2 , then we have a quasi-isomorphismhom F  ♯ (Sym g ( F  )) ( T Y  1 , T − Y  2 ) ≃  CF  ( Y  ) . (1.1)Our main goal is to relate this construction to bordered Heegaard-Floer homology.More precisely, our main results concern the relation between the algebra A ( F  ) introducedin [8] and the Fukaya category of Sym g ( F  ). For 3-manifolds with boundary, we alsopropose (without complete proofs) a dictionary between the A ∞ -module  CFA ( Y  ) of [8]and the generalized Lagrangian submanifold T Y  introduced above. Remark. The cautious reader should be aware of the following issue concerning thechoice of a symplectic form on Sym g ( F  ). We can equip F  with an exact area form, andchoose exact Lagrangian representatives of all the simple closed curves that appear in2  Fukaya categories of symmetric products and bordered Heegaard-Floer homology Heegaard diagrams. By Corollary 7.2 in [12], the symmetric product Sym g ( F  ) carries anexact K¨ahler form for which the relevant product tori are exact Lagrangian. Accordingly, asizeable portion of this paper, namely all the results which do not involve correspondences,can be understood in the exact setting. However, Perutz’s construction of Lagrangiancorrespondences requires the K¨ahler form to be deformed by a negative multiple of thefirst Chern class (cf. Theorem A of [11]). Bubbling is not an issue in any case, becausethe symmetric product of  F  does not contain any closed holomorphic curves (also, wecan arrange for all Lagrangian submanifolds and correspondences to be balanced  and inparticular monotone). Still, we will occasionally need to ensure that our results hold forthe perturbed K¨ahler form on Sym g ( F  ) and not just in the exact case. 1.2. Fukaya categories of symmetric products Let Σ be a double cover of the complex plane branched at n points. In Section 2,we describe the symmetric product Sym k (Σ) as the total space of a Lefschetz fibration f  n,k , for any integer k ∈ { 1 ,...,n } . The fibration f  n,k has  nk  critical points, and theLefschetz thimbles D s ( s ⊆ { 1 ,...,n } , | s | = k ) can be understood explicitly as productsof arcs on Σ.For the purposes of understanding bordered Heegaard-Floer homology, it is natural toapply these considerations to the case of the once punctured genus g surface F  , viewed as adouble cover of the complex plane branched at 2 g +1 points. However, the algebra A ( F,k )considered by Lipshitz, Ozsv´ath and Thurston only has  2 gk  primitive idempotents [8],whereas our Lefschetz fibration has  2 g +1 k  critical points.In Section 3, we consider a somewhat easier case, namely that of a twice puncturedgenus g − 1 surface F  ′ , viewed as a double cover of the complex plane branched at 2 g points. We also introduce a subalgebra A 1 / 2 ( F  ′ ,k ) of  A ( F,k ), consisting of collectionsof Reeb chords on a matched pair  of pointed circles, and show that it has a naturalinterpretation in terms of the Fukaya category of the Lefschetz fibration f  2 g,k as definedby Seidel [15, 16]: Theorem 1.1. A 1 / 2 ( F  ′ ,k ) is isomorphic to the endomorphism algebra of the exceptional collection  { D s , s ⊆ { 1 ,..., 2 g } , | s | = k } in the Fukaya category  F  ( f  2 g,k ) . By work of Seidel [16], the thimbles D s generate the Fukaya category F  ( f  2 g,k ); hencewe obtain a derived equivalence between A 1 / 2 ( F  ′ ,k ) and F  ( f  2 g,k ).Next, in Section 4 we turn to the case of the genus g surface F  , which we now regardas a surface with boundary, and associate a partially wrapped  Fukaya category F  z tothe pair (Sym k ( F  ) , { z }× Sym k − 1 ( F  )) where z is a marked point on the boundary of  F  (see Definition 4.4). Viewing F  ′ as a subsurface of  F  , we specifically consider the samecollection of   2 gk  product Lagrangians D s , s ⊆ { 1 ,..., 2 g } , | s | = k as in Theorem 1.1.Then we have: Theorem 1.2. A ( F,k ) ≃  s,s ′ hom F  z ( D s ,D s ′ ) . 3  Denis Auroux As we will explain in Section 4.4, a similar result also holds when the algebra A ( F,k ) isdefined using a different matching than the one used throughout the paper.Our next result concerns the structure of the A ∞ -category F  z . Theorem 1.3. The partially wrapped Fukaya category  F  z is generated by the  2 gk  objects D s , s ⊆ { 1 ,..., 2 g } , | s | = k . In particular, the natural functor from the category of  A ∞ -modules over  F  z to that of  A ( F,k ) -modules is an equivalence. Moreover, the same result still holds if we enlarge the category F  z to include compactclosed “generalized Lagrangians” (i.e., sequences of Lagrangian correspondences) of thesort that arose in the previous section. Caveat. As we will see in Section 5, this result relies on the existence of a “partialwrapping” (or “acceleration”) A ∞ -functor from the Fukaya category of  f  2 g +1 ,k to F  z ,and requires a detailed understanding of the relations between various flavors of Fukayacategories. This would be best achieved in the context of a more systematic study of partially wrapped Floer theory, as opposed to the ad hoc approach used in this paper(where, in particular, transversality issues are not addressed in full generality). In § 5we sketch a construction of the acceleration functor in our setting, but do not give fulldetails; we also do not show that the functor is well-defined and cohomologically unital.These properties should follow without major difficulty from the techniques introduced byAbouzaid and Seidel, but a careful argument would require a lengthy technical discussionwhich is beyond the scope of this paper; thus, the cautious reader should be warned thatthe proof of Theorem 1.3 given here is not quite complete. 1.3. Yoneda embedding and  CFA Let Y  be a 3-manifold with parameterized boundary ∂Y  ≃ F  ∪ S 1 D 2 . Following [8],the manifold Y  can be described by a bordered Heegaard diagram, i.e. a surface Σ of genus ¯ g ≥ g with one boundary component, carrying: ã ¯ g − g simple closed curves α c 1 ,...,α c ¯ g − g , and 2 g arcs α a 1 ,...,α a 2 g ; ã ¯ g simple closed curves β 1 ,...,β ¯ g ; ã a marked point z ∈ ∂  Σ.As usual, the β -curves determine a product torus T  β = β 1 ×···× β ¯ g inside Sym ¯ g (Σ). As tothe closed α -curves, using Perutz’s construction they determine a Lagrangian correspon-dence T  α from Sym g ( F  ) to Sym ¯ g (Σ) (or, equivalently,¯ T  α from Sym ¯ g (Σ) to Sym g ( F  )).The object T Y  of the extended Fukaya category F  ♯ (Sym g ( F  )) introduced in § 1.1 is thenisomorphic to the formal composition of  T  β and¯ T  α .There is a contravariantYoneda-type A ∞ -functor Y  from the extended Fukaya categoryof Sym g ( F  ) to the category of right A ∞ -modules over A ( F,g ). Indeed, F  ♯ (Sym g ( F  ))can be enlarged into a partially wrapped A ∞ -category F  ♯z by adding to it the samenon-compact objects (products of properly embedded arcs) as in F  z . This allows us toassociate to a generalized Lagrangian L the A ∞ -module Y  ( L ) =  s hom F  ♯z ( L ,D s ) , 4
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