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.