| Let K ⊂ Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology HF + of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the surface is of genus at least three and the mapping class is nontrivial, we find in particular that in the "second-to-highest level" of Spinc structures, this is isomorphic to a free module (over a certain ring) whose rank is equal to the Lefschetz number of the diffeomorphism. After taking a tensor product with Z/2Z , this agrees precisely with the symplectic Floer homology of the diffeomorphism, as calculated by Gautschi. |
