An approximation theorem for equivariant loop spaces in the compact Lie case

Jeffrey Lawrence Caruso and Stefan Waner

Vol. 117 (1985), No. 1, 27–49

DOI: 10.2140/pjm.1985.117.27

Abstract

Let V be a real orthogonal countable-dimensional representation of the Lie group G and denote by Ω^V Σ^V X the space of maps S^V → Σ^V X = X ∧ S^V , where S^V denotes the one-point compactification of V and where X is an arbitrary G-space with stationary basepoint (if V is infinite-dimensional, Ω^V Σ^V X is taken as the natural colimit over spaces indexed on the finite-dimensional submodules of V ). Since G acts on Ω^V Σ^V X by conjugation, the fixed-set (Ω^V Σ^V X)^G is the subspace of G-equivariant maps. We present here an approximation to (Ω^V Σ^V X)^G in the stable case (V large). This approximation will take the form of a space of “configurations” of G-orbits in V .

Mathematical Subject Classification 2000

Primary: 55P47

Secondary: 55P91, 57S99

Milestones

Received: 11 May 1983

Revised: 1 March 1984

Published: 1 March 1985