I will explain the method of Ozsvath and Szabo for calculating the Heegaard Floer homology of a manifold obtained as the zero surgery along a nullhomologous knot. As an application of this technique, I will calculate the Floer homologies of certain mapping tori. It is interesting to compare these to the Seiberg-Witten invariants which are, as we will see, a "strictly" weaker invariant. Another observation we'll make is that the Heegaard Floer homology can be used as an effective invariant for elements of the mapping class group. It can successfully differentiate certain elements of the Torelli subgroup from the identity.