On the structure of finite coverings of compact connected groups (Q860493)

Let \(\omega:\widetilde{\Gamma}\to\Gamma\) be a covering mapping from a path connected space \(\widetilde{\Gamma}\) onto a connected locally path connected topological group \(\Gamma\) with identity \(e\). Then it is known that for any point \(\tilde{e}\in\omega^{-1}(e)\) there exists a unique structure of a topological group on \(\widetilde{\Gamma}\) such that \(\tilde{e}\) is the identity and \(\omega\) is a homomorphism. In this paper, the authors give the same kind of result for a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected group. Note that they do not suppose that the group is locally connected and to prove the result they construct a family of \(k\)-sheeted covering mappings onto Lie groups which approximates a given \(k\)-sheeted covering mapping. Recall that a mapping \(p:X\to Y\) from a topological space \(X\) onto a topological space \(Y\) is called a \textit{\(k\)-sheeted covering mapping} for some \(k\in{\mathbb N}\) if for every point \(y\in Y\), there are a neighborhood \(W\) in \(Y\) and a partition of \(p^{-1}(W)\) into neighborhoods \(V_1, V_2,\ldots, V_k\) in \(X\) such that, for each \(n=1,2,\ldots,k\), the restriction of \(p\) to \(V_n\) is a homeomorphism of \(V_n\) onto \(W\), and \(p\) is called a \textit{finite-sheeted covering mapping} if it is a \(k\)-sheeted covering mapping for some \(k\in{\mathbb N}\).