2014/07/08 by Ghiggini, Paolo, Lisca, Paolo
#57M25 #57N10 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/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 S2× S1's. We also give some applications of our main result, such as a new proof of the result by Birman and Menasco that if the closure of a braid with n strands is the unlink with n components then the braid is trivial.