Calculating the fundamental group of an orbit space

Armstrong, M. A.

doi:10.1090/S0002-9939-1982-0637181-X

Calculating the fundamental group of an orbit space
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Proc. Amer. Math. Soc. 84 (1982), 267-271 Request permission

Abstract:

Suppose $G$ acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space $X$. Let $\overline G$ denote the closure of $G$ in ${\text {Homeo}}(X)$, and $N$ the smallest normal subgroup of $\overline G$ which contains the path component of the identity of $\overline G$ and all those elements of $\overline G$ which have fixed points. We show that ${\pi _1}(X/G)$ is isomorphic to $\overline G /N$ subject to a weak path lifting assumption for the projection $X \to X/\overline G$.

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