No abstract

Communicated by E.M. Friedlander MSC: 20G30 19C20 a b s t r a c tWe formulate a ''correct'' version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory ''homological symbols algebra''. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.

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.

No abstract