An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Anton, Marian

doi:10.1090/S0002-9947-03-03255-0

An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup
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Trans. Amer. Math. Soc. 355 (2003), 2327-2340 Request permission

Abstract:

Conjecturally, for $p$ an odd prime and $R$ a certain ring of $p$-integers, the stable general linear group $GL(R)$ and the étale model for its classifying space have isomorphic mod $p$ cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if $p$ is regular and certain homology classes for $SL_2(R)$ vanish. We check that this criterion is satisfied for $p=3$ as evidence for the conjecture.

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