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Equivariant Eilenberg-MacLane spaces of type 1. (English) Zbl 0752.55007

Given a discrete group \(G\) and a \(G\)-space \(X\), consider its equivariant fundamental groupoid \(\pi^ GX\), i.e. the contravariant functor sending the homogeneous space \(G/H\) to the ordinary fundamental groupoid of the \(H\)-fixed point set of \(X\). Within the appropriate homotopy category of such functors the author establishes a homotopy equivalence \(\mu: \pi^ G(K(\pi^ GX,1))\to\pi^ G X\), where \(K(\pi^ G X,1)\) denotes the classifying space of \(X\). Moreover, \(\mu\) is unique with respect to a certain property. The equivariant Eilenberg-MacLane spaces \(K(\pi^ G X,1)\) were introduced by W. Lück [Manuscr. Math. 58, 67-75 (1987; Zbl 0617.57021)]. The existence of \(\mu\) is claimed there as well. However, the proof contains a gap, which is filled by the present paper.

MSC:

55P91 Equivariant homotopy theory in algebraic topology

Citations:

Zbl 0617.57021
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