Localized stable homotopy of some classifying spaces

Abstract

In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly date from 1975/6 at which time they were proved using my crude stable decomposition of ωn σnX (20) and differential-geometric techniques with the Becker-Gottlieb transfer. Since then more systematic approaches have been developed towards the stable decompositions ((4); (5); (6); (10); (12)) and towards the transfer ((7), I and II; (8)). Actually, the system-atization of the stable decompositions was already latent in (15) if only I had realized it!

Algebraic $K$-theory and etale cohomology

“…Snaith has proved that the inclusion C P°° -> BU(1) -> BU of spaces induces an equivalence of spectra ( [108], or [107])…”

Explicit Brauer induction for symplectic and orthogonal representations

“…Furthermore, as explained in ( [34]; see also [13], [14]), the maps k 0 j k and k 0ĵ k are exponential with respect to the pairings…”

“…On the other hand, in the stable homotopy category, there is a Snaith splitting ( [32], [34]; see also [13], [14]) of the form…”

James Maps, Segal Maps, and the Kahn-Priddy Theorem

“…We gave general splitting theorems there, and a major concern here is the analysis of their multiplicative behavior. In particular, we gave a new proof of Snaith's stable equivalence [22] between fi"2"A and V ?>1 Dq(R", X) for connected spaces X, and we shall prove that our splitting is compatible with products. Let 2°°d enote the suspension functor from spaces to spectra (denoted Qx in [6]).…”