Homological symbols and the Quillen conjecture

TL;DR

This paper proposes a revised version of the Quillen conjecture related to linear group homology, introducing 'homological symbols algebra' and providing evidence and partial proofs for the conjecture.

Contribution

It formulates a corrected conjecture, introduces homological symbols algebra, and proves the Quillen conjecture in degree two for specific cases.

Findings

Evidence supporting the new conjecture.

Proof of the Quillen conjecture in degree two for rank two and prime 5.

Introduction of homological symbols algebra as a new theoretical framework.

Abstract

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.