Comultiplicativity of the Ozsv{á}th-Szab{ó} contact invariant

Abstract

Suppose that $\Sig$ is a surface with boundary and that $g$ and $h$ are diffeomorphisms of $\Sig$ which restrict to the identity on the boundary. Let $Y_g,$ $Y_h$, and $Y_{hg}$ be the 3-manifolds with open book decompositions given by $(S,g)$, $(S,h)$, and $(S,hg)$, respectively. We show that the Ozsv{\'a}th-Szab{\'o} contact invariant is natural under a comultiplication map $\tilde{\mu}:\heeg(-Y_{hg}) \rightarrow \heeg(-Y_{g}) \otimes \heeg(-Y_{h}). $ It follows that if the contact invariants associated to the open books $(\Sig, g)$ and $(\Sig, h)$ are non-zero then the contact invariant associated to the open book $(\Sig, hg)$ is also non-zero. We extend this comultiplication to a map on $\hfp(-\Yhg)$, and as a result we obtain obstructions to the 3-manifold $\Yhg$ being an $L$-space. We also use this to find restrictions on contact structures which are compatible with planar open books.