On open books for nonorientable 3-manifolds

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.