On the topological classification of linear representations

Abstract

GIVEN a compact Lie group, a basic mathematical problem is the classification of its finitedimensional orthogonal representations up to linear equivalence. Recently such representations have been classified up to certain homotopy-theoretic relations much weaker than linear equivalence[l3, 161. In this paper we shall study the classification up to an intermediate relation-topological equivalence; to be precise, two orthogonal representations {V, py} and {W, pw} (ox: G-,0(X) denotes the representation homomorphism) are topologically equivalent if there is a homeomorphism h: V+ W such that pw = hpJ_’ (i.e., V and W are equivariantly homeomorphic). The main result states that topological equivalence agrees with linear equivalence for a large class of compact Lie groups (see Theorem I below together with Corollary 2.3 and (2.4)); in fact, no examples are known for which the classifications differ. The results of this paper were obtained to answer questions posed by R. Lashof and M. Rothenberg; the topological equivalence question for representations arises naturally in their equivariant smoothing theory for topological G-manifolds. Shortly after the first version of this paper was written, the author discovered from [12, $981 that D. Sullivan had worked out the principal part of the main result some time ago but had not written up his conclusions for publication. The main result may be stated as follows: