Homological actions on sutured Floer homology

Abstract

We define the action of the homology group H_1(M,∂M) on the sutured Floer homology SFH(M,γ). It turns out that the contact invariant EH(M,γ,ξ) is usually sent to zero by this action. This fact allows us to refine an earlier result proved by Ghiggini and the author. As a corollary, we classify knots in #^n(S^1×S^2) which have simple knot Floer homology groups: They are essentially the Borromean knots. This answers a question of Ozsváth. In a different direction, we show that the only links in S^3 with simple knot Floer homology groups are the unlinks.