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 nn strands is the unlink with nn components then the braid is trivial.

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

LISCA, PAOLO
2015-01-01

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 nn strands is the unlink with nn components then the braid is trivial.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/586086
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