On some applications of unstable Adams operations to the topology of Kac-Moody groups

Abstract

ABSTRACT. Localized at almost all primes, we describe the structure of differentials in several important spectral sequences that compute the cohomology of classifying spaces of topological Kac-Moody groups. In particular, we show that for all but a finite set of primes, these spectral sequences collapse and that there are no additive extension problems. We also describe some appealing consequences of these results. The main tool is the use of the naturality properties of unstable Adams operations on these classifying spaces.

The rational cohomology groups of the classifying spaces of Kac-Moody groups

“…By Bousfield and Kan [2], there is a multiplicative spectral sequence with E 2 -term given by the derived limits of the functor H * • F which converges to the cohomology of BG(A). This spectral sequence is used in Kitchloo [13] to study the cohomology of the classifying spaces of Kac-Moody groups. But the E 2 -term of Bousfield-Kan spectral sequence is hard to compute.…”

“…Proof: By Kitchloo [13], let l be a prime number so that Weyl group W (A) contains no elements of order l, then there is an unstable Adams map ψ on BG(A), and it induces the Adams map ψ I on BG I (A) for all I ∈ C(A). The cohomology endomorphism ψ * I acting on H 2q (BG I (A)) for G I (A) of finite type is the multiplication by l q .…”