Homological symbols and the Quillen conjecture

We 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.