arXiv: 1407.2148

Abstract

Let $M$ be a closed, oriented, connected $3$–manifold and $\left(B,\pi \right)$ an open book decomposition on $M$ with page $\Sigma$ and monodromy $\phi$. 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 $\phi$ is trivial and $M$ is a connected sum of copies of $S^2\times S^1$. 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.

Keywords

Mathematical Subject Classification 2010

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