Connected components of Isom(varvec{pmb {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. The properties of the ‘square map’ and other related maps serve as a technical step towards the counting of the connected components of the variety of representations of pi _1(N_k) in Isom(mathbb {H}^3). We show that the variety of representations hom(pi _1(N_k),mathrm {Isom}(mathbb {H}^3)) has 2^{k+1} connected components, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.