Volume 11, issue 1 (2011)

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Knot Floer homology and rational surgeries

Peter S Ozsváth and Zoltán Szabó

Algebraic & Geometric Topology 11 (2011) 1–68
Abstract

Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot $K$. As an application, we express the Heegaard Floer homology of rational surgeries on $Y$ along a null-homologous knot $K$ in terms of the filtered homotopy type of the knot invariant for $K$. This has applications to Dehn surgery problems for knots in ${S}^{3}$. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold with even first Betti number.

Keywords
Floer homology, Dehn surgery
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M27, 57M25