On open books for nonorientable 3-manifolds

Abstract

We show that the monodromy of Klassen's genus two open book for P-2 x S-1 is the Y-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that S-2 (x) over tilde S-1 admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of P(2)xS(1) and S-2 (x) over tilde 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 x S-1 and S-2 (x) over tilde 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.