Connected components of Isom(${\mathbb{H}}^3$)-representations of non-orientable surfaces

Abstract

Let $N_k$ denote the closed non-orientable surface of genus $k$. In this paper we study the behaviour of the `square map' from the group of isometries of hyperbolic 3-space to the subgroup of orientation preserving isometries. We show that there are $2^{k+1}$ connected components of representations of ${\pi }_1(N_k)$ in Isom$({\mathbb{H}}^3)$, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.

13 pages