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# Mathematical Research Letters

## Volume 21 (2014)

### Number 5

### Homological actions on sutured Floer homology

Pages: 1177 – 1197

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n5.a12

#### Author

#### Abstract

We define the action of the homology group $H_1 (M, \partial M)$ 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. 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.