Homological actions on sutured Floer homology

We define the action of the homology group $H_1(M,\mathrm{\partial }M)$ on the sutured Floer homology $SFH(M,\gamma )$. It turns out that the contact invariant $EH(M,\gamma ,\xi )$ 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\times S^2) which have simple knot Floer homology groups: They are essentially the Borromean knots.