  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.
