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

Abstract

AbstractLet $$N_k$$ 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 $$\pi _1(N_k)$$ π 1 ( N k ) in Isom$$(\mathbb {H}^3)$$ ( H 3 ) . We show that the variety of representations hom($$\pi _1(N_k),\mathrm {Isom}(\mathbb {H}^3)$$ π 1 ( N k ) , Isom ( H 3 ) ) has $$2^{k+1}$$ 2 k + 1 connected components, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.