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 $S^2\tilde{\times }{S}^1$ 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 $S^2\tilde{\times }{S}^1$. Finally, we formulate a version of Gabai's theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.

Final version, to appear in Periodica Mathematica Hungarica

References in corpus (2)

• Lectures on open book decompositions and contact structures
• Generalized plumbings and Murasugi sums