Open book decompositions versus prime factorizations of closed, oriented 3-manifolds

Let $M$ be a closed, oriented, connected 3--manifold and $(B,\pi)$ an open book decomposition on $M$ with page $\Sigma$ and monodromy $\varphi$. It is easy to see that the first Betti number of $\Sigma$ is bounded below by the number of $S^2\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $\varphi$ is trivial and $M$ is a connected sum of $S^2\times S^1$'s. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with $n$ strands is the unlink with $n$ components then the braid is trivial.Comment: 8 pages, 1 figure. Submitted to the proceedings of the conference "Interactions between low dimensional topology and mapping class groups", July 1-5, 2013, Max Planck Institute for Mathematics, Bon

Infinitely many universally tight contact manifolds with trivial Ozsvath-Szabo contact invariants

In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.Comment: This is the version published by Geometry & Topology on 2 April 200

Embedded contact homology and open book decompositions

This is the first of a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology (abbreviated ECH). In this paper we prove that, given a closed, oriented, contact $3$-manifold, there is an equivalence between ECH of the closed $3$-manifold and a version of ECH, defined on the complement of the binding of an adapted open book decomposition.Comment: First revision: A mistake in our use of Morse-Bott techniques in ECH has been fixed and results on sutured ECH have been added. Second revision: bibliography updated. Third revision: exposition improved and details added following referees' suggestions. Fourth revision: we added an appendix sketching the proof of the Morse-Bott gluing statements used in the articl