闭定向三维流形的开书分解与素因子分解
Open book decompositions versus prime factorizations of closed, oriented 3-manifolds
摘要：
设$M$是一个闭的、定向的、连通的3-流形，$(B,π)$是$M$上的一个开书分解，其页为$Σ$，单值为$φ$ .很容易看出，$Σ$的第一个贝蒂数由$S^2\times S^1$的个数限定，即$M$的素因子分解中的因子。我们的主要结果是等式实现的充要条件是$φ$是平凡的且$M$是$S^2\times S^1$的连通和.我们也给出了我们的主要结果的一些应用，如Birman和Menasco的结果的一个新的证明，如果一个辫子的封闭与$n$股是unlink与$n$组件，那么辫子是平凡的。
英文摘要：
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 $S^2\times S^1$--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 $S^2\times S^1$'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 $n$ components then the braid is trivial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
