Calculating the fundamental group of an orbit space

Abstract

Suppose G G acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space X X . Let G ¯ \overline G denote the closure of G G in Homeo ( X ) {\text {Homeo}}(X) , and N N the smallest normal subgroup of G ¯ \overline G which contains the path component of the identity of G ¯ \overline G and all those elements of G ¯ \overline G which have fixed points. We show that π 1 ( X / G ) {\pi _1}(X/G) is isomorphic to G ¯ / N \overline G /N subject to a weak path lifting assumption for the projection X → X / G ¯ X \to X/\overline G .