Toward conjectures of Rognes and Church--Farb--Putman (Lecture Remotely)
Wednesday, August 3, 2022 - 9:00am - 10:00am
In this talk I will give an overview of two related projects. The first project concerns the high-degree rational cohomology of the special linear group of a number ring R. Church--Farb--Putman conjectured that, when R is the integers, these cohomology groups vanish in a range close to the virtual cohomological dimension. The groups SL_n(R) satisfy a twisted analogue of Poincare duality called virtual Bieri--Eckmann duality, and their rational cohomology groups are governed by SL_n(R)-representations called the Steinberg modules. I will discuss a recent proof of the codimension two case of the Church--Farb--Putman conjecture using the topology of certain simplicial complexes related to the Steinberg modules. The second project concerns Rognes’ connectivity conjecture on a family of simplicial complexes (the common basis complexes) with implications for algebraic K-theory. I will describe work-in-progress proving Rognes' conjecture for fields, and its connections to SL_n(R) and the Steinberg modules. This talk includes past and ongoing work joint with Benjamin Brück, Alexander Kupers, Jeremy Miller, Peter Patzt, Andrew Putman, and Robin Sroka.