# Comultiplication and the Ozsvath-Szabo contact invariant

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John Baldwin, Columbia University
Fine Hall 314

Let $S$ be a surface with boundary and suppose that $g$ and $h$ are diffeomorphisms of $S$ which restrict to the identity on the boundary. I'll describe how the Ozsvath-Szabo contact invariants associated to the open books $(S,g)$, $(S,h)$, and $(S,hg)$ are natural with respect to a comultiplication on the corresponding Heegaard-Floer homology groups. In particular, it follows that if the contact invariants associated to the open books $(S,g)$ and $(S,h)$ are non-zero, then so is the contact invariant associated to the open book $(S,hg)$. I plan to discuss an extension of this comultiplication to ${HF}^+$ and an obstruction to the compatibility of a contact structure with a planar open book.