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

TL;DR

This paper establishes a precise relationship between open book decompositions and prime factorizations of closed, oriented 3-manifolds, characterizing when the Betti number equals the number of $S^2 imes S^1$ factors and applying this to braid theory.

Contribution

It proves that equality of the Betti number and the number of $S^2 imes S^1$ factors occurs only when the monodromy is trivial and the manifold is a connected sum of $S^2 imes S^1$'s, providing new insights into 3-manifold topology.

Findings

Equality of Betti number and prime factors occurs iff monodromy is trivial

Characterization of manifolds with minimal Betti number in open book decompositions

New proof of a braid triviality result by Birman and Menasco

Abstract

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.