#### Volume 19 (2015)

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

### Paolo Ghiggini and Paolo Lisca

Geometry & Topology Monographs 19 (2015) 145–155
 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 ${S}^{2}×{S}^{1}$–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 ${S}^{2}×{S}^{1}$. 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
open book decomposition, prime factorization, $3$–manifold
Primary: 57N10
Secondary: 57M25