{"paper":{"title":"Open book decompositions versus prime factorizations of closed, oriented 3-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Paolo Ghiggini, Paolo Lisca","submitted_at":"2014-07-08T16:06:41Z","abstract_excerpt":"Let $M$ be a closed, oriented, connected 3--manifold and $(B,\\pi)$ an open book decomposition on $M$ with page $\\Sigma$ and monodromy $\\varphi$. It is easy to see that the first Betti number of $\\Sigma$ is bounded below by the number of $S^2\\times S^1$--factors in the prime factorization of $M$. Our main result is that equality is realized if and only if $\\varphi$ is trivial and $M$ is a connected sum of $S^2\\times S^1$'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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2148","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}