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28 March 2024
 
  » arxiv » 2004.8173

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On open books for nonorientable 3-manifolds
Burak Ozbagci ;
Date 17 Apr 2020
AbstractWe show that the monodromy of Klassen’s genus two open book for $P^2 imes S^1$ is the $Y$-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that $S^2 widetilde{ imes} S^1$ admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of $P^2 imes S^1$ and $S^2 widetilde{ imes} S^1$ 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 $P^2 imes S^1$ and $S^2 widetilde{ imes} S^1$. Finally, we formulate a version of Gabai’s theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.
Source arXiv, 2004.8173
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