Contact monoids and Stein cobordisms

Suppose $S$ is a compact surface with boundary, and let $\phi$ be a diffeomorphism of $S$ which fixes the boundarypointwise. We denote by $(M_{S,\phi},\xi_{S,\phi})$ the contact three-manifold compatible with the open book $(S,\phi)$.In this paper, we construct a Stein cobordism from the contact connected sum $(M_{S,h},\xi_{S,h})\,\#\,(M_{S,g},\xi_{S,g})$to $(M_{S,hg},\xi_{S,hg})$. This cobordism accounts for the comultiplication map on Heegaard Floer homology discoveredin~\cite{bald3}, and illuminates several geometrically interesting monoids in the mapping class group $\mathrm{Mod}^+(S,\partial S)$. We derive some consequences for the fillability of contact manifolds obtained as cyclic branched coversof transverse knots.

Full Text (PDF format)

Published 2 May 2012