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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


Let M be a closed, oriented, connected 3–manifold and (B,π) an open book decomposition on M with page Σ and monodromy φ. It is easy to see that the first Betti number of Σ is bounded below by the number of S2 × S1–factors in the prime factorization of M. Our main result is that equality is realized if and only if φ is trivial and M is a connected sum of copies of S2 × S1. 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.

open book decomposition, prime factorization, $3$–manifold
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57M25
Received: 18 November 2014
Accepted: 18 November 2014
Published: 29 December 2015
Paolo Ghiggini
Laboratoire de Mathématiques Jean Leray
Université de Nantes & CNRS
BP 92208
2 rue de la Houssinière
F-44322 Nantes 03
Paolo Lisca
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
I-56127 Pisa