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SELF-DIFFEOMORPHISMS OF THREE-MANIFOLDS AND HEEGAARD FLOER HOMOLOGY

An orientation preserving diffeomorphism acting on an oriented, closed 3manifold has a predictable effect on Heegaard Floer homology groups. Although this may be well-known to experts in the field, details of a proof have not appeared in the literature. The purpose of this paper is to give an explicit proof that Heegaard Floer homology groups remain invariant under an action on the set of spin structures induced by a self-diffeomorphism of the manifold. Our proof is based on the definitions of these groups and the result gives computable obstructions to the existence of certain group actions on manifolds. We illustrate this through an example where a given 3-manifold is obstructed from being a q-fold cyclic cover of the 3-sphere with branching set a knot.