E∞-cells and general linear groups of infinite fields

Abstract

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of E∞-algebras. We prove that there is a vanishing line of slope 2 for their E∞-homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others