Decorated Heegaard Diagrams and Combinatorial Heegaard Floer Homology
Open AccessHeegaard Floer homology is a collection of invariants for closed oriented three-manifolds, introduced by Ozsvath and Szabo in 2001. The simplest version is defined as the homology of a chain complex coming from a Heegaard diagram of the three manifold. In the original definition, the differentials count the number of points in certain moduli spaces of holomorphic disks, which are hard to compute in general.More recently, Sarkar and Wang (2006) and Ozsvath, Stipsicz and Szabo, (2009) have determined combinatorial methods for computing this homology with Z_2 coefficients. Both methods rely on the construction of very specific Heegaard diagrams for the manifold, which are generally very complicated.Given a decorated Heegaard diagram H for a closed oriented 3-manifold Y, that is a Heegaard diagram together with a collection of embedded paths satisfying certain criteria, we describe a combinatorial recipe for a chain complex CF'(H). If H satisfies some technical constraints we show that this chain complex is homotopically equivalent to the Heegaard Floer chain complex CF(H) and hence has the Heegaard Floer homology HF(Y) as its homology groups. Using branched spines we give an algorithm to construct a decorated Heegaard diagram which satisfies the necessary technical constraints for every closed oriented Y. We present this diagram graphically in the form of a strip diagram.
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Hammarsten_gwu_0075A_12732.pdf | 2018-01-16 | Open Access |
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