A cylindrical reformulation of Heegaard Floer homology

Abstract

A cylindrical reformulation of Heegaard Floer homology ROBERT LIPSHITZWe reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold † OE0; 1 R, where † is the Heegaard surface, instead of Sym g . †/. We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the @-operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsváth-Szabó. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our construction, as well as a brief discussion of the relation with a program of C Taubes. 57R17; 57R58, 57M27In [21], P Ozsváth and Z Szabó associated to a three-manifold Y and a Spin ‫ރ‬ -structure s on Y a collection of abelian groups, known together as Heegaard Floer homology. These groups, which are believed to be isomorphic to certain Seiberg-Witten Floer homology groups and ), fit into the framework of a .3 C 1/-dimensional topological quantum field theory. Since its discovery around the turn of the millennium, Heegaard Floer homology has been applied by Ozsváth, Rasmussen and Szabó to the study of knots and surgery [19; 25; 18], contact structures [23] and symplectic structures [22], and is strong enough to reprove most results about smooth four-manifolds originally proved by gauge theory [17]. In this paper we give an alternate definition of the Heegaard Floer homology groups.Rather than being associated directly to a three-manifold Y , the Heegaard Floer homology groups defined in [21] and in this paper are associated to a Heegaard diagram for Y , as well as a Spin ‫ރ‬ -structure s and some additional structure. A Heegaard diagram is a closed, orientable surface † of genus g , together with two g -tuples of pairwise disjoint, homologically linearly independent, simple closed curves Ę D f˛1;;˛gg and Ě D fˇ1; ;ˇgg in †. A Heegaard diagram specifies a threemanifold as follows. Thicken † to † OE0; 1. Glue thickened disks along the˛i f0g and along theˇj f1g. The resulting space has two boundary components, each homeomorphic to S 2 . Cap each with a three-ball. The result is the three-manifold specified by . †; Ę; Ě /. Different Heegaard diagrams can specify the same three-manifold. Two different Heegaard diagrams specify the same three-manifold if and only if they agree after a sequence of moves of the following three kinds:Isotopies of the˛-orˇ-circles.Handleslides among the˛-orˇ-circles. These correspond to pulling one˛-(orˇ-) circle over another.Stabilization, which corresponds to taking the connect sum of the Heegaard diagram with the standard genus-one Heegaard diagram for S 3 .See Gompf and Stipsicz [9, Sections 4.3 and 5.1] or Ozsváth and Szabó [21, Section 2] for more details.So, after associating the Heegaard Floer homology groups to a Heegaard diagram, one must prove they are unchanged by these three kinds of Heegaard moves (as well as deforming the additiona...

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