| Floer homology provides a lower bound on the number of fixed points of symplectomorphisms in a given mapping class, in a suitable sense. In the first part of this thesis, we calculate the perturbed Heegaard Floer homology for the product manifold S1 x Sigma g and for some special classes of fibered three-manifolds, including the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves in the second highest Spin c structure. Along the way, we also establish an adjunction inequality for the perturbed Heegaard Floer homology, which indicates a potential connection between the U-action on the homology group and the Thurston norm of a three-manifold.The second part of this thesis is devoted to an application of Heegaard Floer homology to Dehn surgery, for which we resolve some special cases of the cosmetic surgery conjecture in S3. More precisely, let K be a non-trivial knot in S 3, and let r and r' be two distinct rational numbers of the same sign we prove that there is no orientation-preserving homeomorphism between the manifolds S3r (K) and S3r' (K). We further generalize this uniqueness result to knots in arbitrary integral homology L-spaces. |
