| We compute the Heegaard Floer homology, as an absolutely graded module, of the result Yk,q of any –1/ q surgery on the k-fold connected sum of Borromean knots. We analyze certain infinite families Fn of homeomorphic Stein fillings of fixed contact 3-manifolds first constructed by Akhmedov, Etnyre, Mark and Smith. For each n, the boundary of the Stein fillings in Fn is one of the manifolds Yk,2. We then compute the relative monopole 4-manifold invariants of each member of Fn, using the recent result of Kutluhan, Lee, and Taubes that Heegaard Floer homology is isomorphic to a version of monopole Floer homology. The main result is that these relative invariants distinguish the members of Fn up to diffeomorphism. We also analyze the relative invariants in Heegaard Floer homology. We then give some new constructions of Stein fillings. We obtain an infinite family of Stein fillings, homeomorphic but pairwise non-diffeomorphic, which all embed into a common Stein manifold. We construct an infinite family of Stein fillings which are pairwise non-homeomorphic whose boundary has b1 = 1, and find contact 3-manifolds with b 1 = 0 which admit arbitrarily many non-homeomorphic Stein fillings. We conclude with a computation of two examples of maps on Heegaard Floer homology induced by the cobordism obtained by attaching a new critical fiber to a Lefschetz fibration. In an appendix, we present a computation of the Heegaard Floer homology of –1/q surgery on the k-fold connected sum of trefoil knots. |
