Equivariant covering spaces and homotopy covering spaces
The theory of covering spaces is very closely related to the fundamental groupoid in the nonequivariant case for a non-connected base space, and so it is also equivariantly, in the same form as it relates to the fundamental group in the non-equivariant case with connected base space. This paper starts by defining the fundamental grupoid \(\Pi_GX\) following tom Dieck [\textit{T. tom Dieck}, Transformation groups. Studies in Math., vol. 8, (Berlin-New York:) Walter de Gruyter, (1987; Zbl 0611.57002)], as the category whose objets are the equivariant maps \(x:G/H\rightarrow X\) and whose morphisms are the pairs \((\omega,\alpha)\) where \(\alpha:G/H\rightarrow G/K\) is a \(G\)-map and \(\omega\) is an equivalence class of paths \(x\rightarrow y\circ\alpha\) in \(X^H\). The paper is constructed on the basis of proving analogous results to those one finds in the non-equivariant situation. Let us denote by \({\mathcal O}_G\) the orbit category of \(G\) whose objects are the homogeneous spaces \(G/H\) and whose morphisms are \(G\)-maps. The category of the homotopy \(G\)-covering spaces over orbits, that is of constructions of the form \(G\times_HF\) and \(G\)-fiber homotopy equivalences is denoted by \({\mathcal D}_G\), and \(h{\mathcal D}_G\) is the homotopy category over \({\mathcal O}_G\) of \({\mathcal D}_G\). There is a projection map \(\pi:{\mathcal D}_G\rightarrow{\mathcal O}_G\) taking \(G\times_HF\) to \(G/H\). There are two main results in this paper, and they are condensed in the following one. \textbf{Theorem.} Given a \(G\)-space \(X\) of the \(G\)-homotopy type of a \(G\)-\(CW\) complex, \textbf{1.} the equivalence classes of \(G\)-covering spaces over \(X\) are in 1-1 correspondence with the isomorphism classes of discrete representations of the fundamental groupoid; \textbf{2.} the (weak) fiber-homotopy equivalence classes of (weak) homotopy \(G\)-covering spaces are in 1-1 correspondence with the isomorphism classes of homotopy discrete representations of the fundamental groupoid; \textbf{3.} thus the classifications are equivalent problems (see [\textit{J. P. May}, Trans. Am. Math. Soc. 258, 127--146 (1980; Zbl 0429.55004)]); \textbf{4.} the equivalence classes of \(G\)-covering spaces over \(X\) are in 1-1 correspondence with the equivalence classes of strictly discrete \(\Pi_GX\)-actions; and \textbf{5.} the (weak) fiber-homotopy equivalence classes of (weak) homotopy \(G\)-covering spaces over \(X\) are in 1-1 correspondence with the equivalence classes of discrete \(\Pi_GX\)-actions. The proof is based upon generalizing a construction due to Elmendorf [\textit{A. D. Elmendorf}, Trans. Am. Math. Soc. 277, 275-284 (1983; Zbl 0521.57027)] and passes through the proof that certain functors \(\pi_0:{\mathcal D}_G\rightarrow{\mathcal F}_G\) and \(\pi:h{\mathcal D}_G\rightarrow{\mathcal F}_G\) are both continuous and weak equivalences on mapping spaces, where \({\mathcal F}_G\) is the category whose objects are the pairs \((G/H,\alpha)\) for \(\alpha:{\mathcal O}_G\downarrow(G/H)\rightarrow{\text \mathbf {Set}}\) being a contravariant functor. This paper is also full of illustrating examples for both the finite and the compact Lie case of positive dimension.
scientific article; zbMATH DE number 2153121