Contact monoids and Stein cobordisms

Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},\xi_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, 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}), for any two boundary-fixing diffeomorphisms h and g. This cobordism accounts for the comultiplication map on Heegaard Floer homology discovered in an earlier paper by the author, and it illuminates several geometrically interesting monoids in the mapping class group of S. We derive some consequences for the fillability of contact manifolds obtained as cyclic branched covers of transverse knots.