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On open books for nonorientable 3-manifolds

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Abstract

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.

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  1. Department of Mathematics, Koç University, Istanbul, Turkey

    Burak Ozbagci

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  1. Burak Ozbagci
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Ozbagci, B. On open books for nonorientable 3-manifolds. Period Math Hung 84, 56–69 (2022). https://doi.org/10.1007/s10998-021-00393-y

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