Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2021-06-29 , DOI: 10.1007/s10998-021-00393-y Burak Ozbagci
We show that the monodromy of Klassen’s genus two open book for \(P^2 \times S^1\) is the Y-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that \(S^2 \mathbin {\widetilde{\times }}S^1\) admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of \(P^2 \times S^1\) and \(S^2 \mathbin {\widetilde{\times }}S^1\) admits infinitely many isomorphic genus two open books whose monodromies are mutually nonisotopic. Furthermore, we include a simple observation about the stable equivalence classes of open books for \(P^2 \times S^1\) and \(S^2 \mathbin {\widetilde{\times }}S^1\). Finally, we formulate a version of Stallings’ theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.
中文翻译:
关于不可定向三流形的开卷
我们证明 Klassen 的 genus two open book for \(P^2 \times S^1\)的单向性是 Lickorish的Y 同胚,也称为 crosscap slide。类似地,我们证明\(S^2 \mathbin {\widetilde{\times }}S^1\)承认一个 genus 2 open book,其单一性是 crosscap 转置。此外,我们证明\(P^2 \times S^1\)和\(S^2 \mathbin {\widetilde{\times }}S^1\) 中的每一个都承认无限多的同构属 两个打开的书,其单倍性是相互非同位素的。此外,我们对\(P^2 \times S^1\)和\(S^2 \mathbin {\widetilde{\times }}S^1\)的开放书籍的稳定等价类进行了简单的观察. 最后,我们制定了关于打开书籍的 Murasugi 和的 Stallings 定理的一个版本,而不在页面上强加任何定向性假设。