S^equivariant Function Spaces and Characteristic Classes

We determine the structure of the homology of the BeckerSchultz space SGiS1) ~ Q(CP^° AS1) of stable Sx-equivariant self-maps of spheres (with standard free S1 -action) as a Hopf algebra over the DyerLashof algebra. We use this to compute the homology of BSGiS1). Along the way, we give a fresh account of the partially framed transfer construction and the Becker-Schultz homotopy equivalence. We compute the effect in homology of the '^-transfers" CPf AS1^ Q((BZpn) + ), n > 0, and of the equivariant J-homomorphisms SO -> Q(RP£°) and U -» Q(CP°° A S1). By composing, we obtain U —> QS° in homology, answering a question of J. P. May. Introduction. Let ii be a compact Lie group admitting a finite-dimensional orthogonal representation W such that H acts freely on the unit sphere sW. H must thus be S1, S3, the normalizer of S1 in S3, or one of a known list [13] of finite groups with periodic cohomology, including (as subgroups of S3) the cyclic and generalized quaternion groups. Let EndB(sW) denote the space of ü-equivariant continuous self-maps of sW. By joining with the identity map we obtain inclusions EndH(s(nW)) C EndH(s((n+l)W)), and we write G(H) for the direct limit. The homotopy type of G(H) was determined by J. C. Becker and R. E. Schultz [2], and turns out to be independent of W. If we write SG(H) for the component of G(H) containing the identity map (so SG(H) = G(H) if H is connected), then in [3], Becker and Schultz (see also [9] in case H is finite) enrich the composition product in SG(H) to an infinite-loop space structure. The classifying space BSG(H) classifies oriented spehrical fibrations with a fiber-preserving ü-action modelled on s(nW), stabilized by forming fiberwise joins with the trivial if-fibration with fiber sW. In this paper we determine the modp homology of SG(S1) and of BSG(S1) as Hopf algebras over the Dyer-Lashof algebra. Along the way, we compute the effect in homology of the "forgetful" maps SG(S1) —> 5G(Zpn) and of the equivariant J-homomorphisms Jz, : SO -» SG(Z2) and jSi : U -+ SG(S1). The starting point for our analysis is the study of certain "transfer" maps. §1 is devoted to an account of the construction and general properties of these maps. In §2 we study certain transfers t associated to an inclusion K C H of compact Lie groups. If E is a smooth principal H-space, then t:(E/HY" ^Q((E/Ky«), where ÇB is the vector-bundle obtained by mixing E J, E/H with the adjoint representation of H on its Lie algebra, the superscript denotes formation of the Thom space and QX is the enveloping infinite loop space of X. Received by the editors August 12, 1980 and, in revised form, June 15, 1985. 1980 Mathematics Subject Classification. Primary 55R40; Secondary 55R91, 55Q50, 55R12. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page