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

Let M be a closed, oriented, connected 3–manifold and .B; / an open book decomposition on M with page † and monodromy ' . It is easy to see that the first Betti number of † is bounded below by the number of S S –factors in the prime factorization of M . Our main result is that equality is realized if and only if ' is trivial and M is a connected sum of copies of S S . We also give some applications of our main result, such as a new proof of the fact that if the closure of a braid with n strands is the unlink with n components then the braid is trivial.