Capping off open books and the Ozsváth–Szabó contact invariant

If $(S,\phi)$ is an open book with disconnected binding, then we can form a new open book $(S',\phi')$ by capping off one of the boundary components of $S$ with a disc. Let $M_{S,\phi}$ denote the 3-manifold with open book decomposition $(S,\phi)$. We show that there is a $U$-equivariant map from ${HF^+}(-M_{S',\phi'})$ to ${HF^+}(-M_{S,\phi})$ which sends $c^+(S',\phi')$ to $c^+(S,\phi)$, and we discuss various applications. In particular, we determine the support genera of almost all contact structures that are compatible with genus one, one boundary component open books. In addition, we compute $d_3(\xi)$ for every tight contact manifold $(M,\xi)$ supported by a genus one open book with periodic monodromy.

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Published 27 September 2013