$E_{\infty }$-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 $E_{\infty }$-algebras. We prove that there is a vanishing line of slope 2 for their $E_{\infty }$-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.

82 pages; final version to appear in Duke Mathematical Journal