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 $\mathrm{\Sigma }$ and monodromy $\phi$. It is easy to see that the first Betti number of $\mathrm{\Sigma }$ 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 $\phi$ 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.