## 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

Zbl 0617.57021
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