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.Comment: 12 pages, 5 figure

Comultiplication in link Floer homology and transversely non-simple links

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard generators of B_n. This gives rise to a "comultiplication" map on link Floer homology. We use this to generate infinitely many new examples of prime topological link types which are not transversely simple.Comment: 16 pages, 10 figure