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

Abstract

Abstract. Conjecturally, for p an odd prime and R a certain ring of pintegers, the stable general linear group GL(R) and theétale 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.

Unstable analogues of the Lichtenbaum-Quillen conjecture

“…The case of each prime would have to be dealt with separately and the computational complexity would grow rapidly with the prime. As a consequence of Theorem 9 we have the following result [3]:…”

“…In [3] the prime ℓ and the ring A are as in [2]. The focus is on analyzing the image I n of H * (BGL n (A); F ℓ ) in H * (BD n (A); F ℓ ) with respect to the restriction homomorphism res n associated to the inclusion of the subgroup D n (A) of diagonal matrices into GL n (A).…”

Generators for Group Homology and a Vanishing Conjecture

“…Proof. We first note that R is a Euclidean ring [4], which, by Lemma 7.2 [2] implies that SL 2 (R) is a perfect group. Thus, applying the spectral sequence 2.3 to the extension…”

“…is surjective. Anton's reformulation of Quillen's conjecture in [3] and results in [2] imply that map 2.10 factorizes thusly:…”