E_∞-cells and general linear groups of infinite fields

2020/05/12 by Galatius, Soren, Kupers, Alexander, Randal-Williams, Oscar
#18F25 #20G15 #55P48 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2005.05620

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.

Related

• E_∞-cells and general linear groups of finite fields
• Euler class groups, and the homology of elementary and special linear groups
• Reidemeister spectrum of special and general linear groups over some\n fields contains 1
• The category of E_∞-coalgebras, the E_∞-coalgebra structure on the homology, and the dimension completion of the fundamental group
• Appoximate Cohomology