Volume 19 (2015)

Download this article
Download this article For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
Purchases
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
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 (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.

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