On open books for nonorientable 3-manifolds

We show that the monodromy of Klassen’s genus two open book for $P^2 \times S^1$isthe$Y$-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that $S^2\tilde{\times }{S}^1$ admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of $P^2\times S^1$ and $S^2\tilde{\times }{S}^1$ 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 $S^2 \widetilde{\times} S^1$. Finally, we formulate a version of Gabai’s theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.