On lifting transformation groups

Abstract

It is known [1] that if (X, G, IA) is a topological transformation group such that 7r1(G) = 0, and if X is a covering space of X, then there is a unique topological transformation group (5, G, ,) which covers (X, G, IA). In [2] the fundamental group o(X, xo, G) of a group of homeomorphisms G of a topological space X is defined,, and, it is observed in the proof of Theorem 7 of that paper that o-(X, xo, G) acts in a natural way as a group of homeomorphisms of the universal covering space X. In this note the relationship between these two results is investigated, and equations defining A are found. The language and notations will be minor modifications of those of [2], together with standard notations for covering spaces. It will be assumed throughout that X is path-connected, locally path-connected, and locally simply connected, that G is a locally pathconnected topological group, and that (X, G, IA) is a topological transformation group. First observe that every homeomorphism g of a path-connected space X induces an automorphism of the group N of normal subgroups of 7ri(X, xo). If k is a path from gx0 to xo, then the map [f]-4[kp+gf+k] is an automorphism of xri(X, x0). The image of a normal subgroup lro is a normal subgroup which is independent of k, and which will be denoted by g*iro. It is easily checked that g*: N-*N is an automorphism. DEFINITION 1. A normal subgroup ir0 of 7r,(X, x0) is said to be invariant under G if for every gEG, g*ro i=r0. Let [f; g]w, denote the equivalence class of the path f of order g under the equivalence relation