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

P. Ghiggini, P. Lisca
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引用次数: 2

Abstract

Let M be a closed, oriented, connected 3–manifold and .B; / an open book decomposition on M with page † and monodromy ' . It is easy to see that the first Betti number of † is bounded below by the number of S S –factors in the prime factorization of M . Our main result is that equality is realized if and only if ' is trivial and M is a connected sum of copies of S S . 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.
开卷分解与封闭定向3流形的质因数分解
设M是一个封闭的、定向的、连通的3流形,b;/一个打开的书在M上的分解,页面为†,单字为'。很容易看出,†的第一个Betti数以M的质因数分解中S个S因子的个数为界。我们的主要结论是当且仅当'是平凡的且M是S S的拷贝的连通和时,等式才得以实现。我们还给出了主要结果的一些应用,例如一个新的证明,如果一个n股辫状体的闭包是有n个分量的不连接,那么这个辫状体是平凡的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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