Fibered representations, an open condition (Q1304858)

Let \(M\) be a closed connected smooth manifold with finitely generated discrete fundamental group \(\pi_1(M) = \Gamma\), let \(G\) be a Lie group with finitely many connected components, and let \(K\) be a maximal compact subgroup of \(G\). The main result, which generalizes previous work of \textit{D. Tischler} [Topology 9, 153-154 (1970; Zbl 0177.52103)], shows that the set of all representations \(\varphi: \Gamma \rightarrow G\), such that the image of \(\varphi\) is discrete and torsion free, contains an open subset consisting of all \(\varphi\), which are represented by smooth fibrations, i.e., for which there exists a smooth fibre bundle projection \(q: M \rightarrow \varphi(\Gamma)\backslash G\slash K\), such that \(q_{\#}: \Gamma \rightarrow \pi_1(N)\) is equal to \(\varphi\).