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Lifting Group Actions in Fibre Bundles

1961; Princeton University; Volume: 74; Issue: 1 Linguagem: Inglês

10.2307/1970310

1939-8980

Thomas E. Stewart,

Advanced Operator Algebra Research

In studying the actions of a compact group on a topological space X, one very soon arrives at the conclusion that the smaller the codimension of the largest orbit of the action the more manageable the problem becomes. For example, if G acts on euclidean n-space with an orbit of dimension greater than n 3, then G essentially acts linearly [5]. Similarly if G is a simply connected Lie group acting on itself transitively, then the action is essentially left translation. Unfortunately, this at least demands that the dimension of G itself be relatively large with respect to that of X. However, these results suggest that in order to study the action of an arbitrary compact group H on X, we might attempt to imbed this action in an action of a larger compact group G on X hence conceivably raising the dimension of the orbits. In this paper we will be concerned with the following type of problem. If H is an invariant subgroup of G' and we are given an action of G' on X, then we have associated canonically an action of G'/H on the orbit space X/H [6, p. 61]. When is the converse true? That is, given H acting on X and G'/H acting on the orbit space X/H when can we obtain a consistent action of G' on X? We will restrict ourselves throughout to the case where H acts freely on X and G' = G x H, though many of the theorems can be proved in the more general setting of [8]. With H acting freely, we arrive at the situation of a principal fibre space in the sense of Cartan. In ? 2 we study the case where G = G'/H acts transitively on the orbit space B = X/H which is then the base space of this fibre space.