Bonn Topology Group - Abstracts

General Information - Members - Activities - Topology Seminar

Talk

April 21st 2020
Søren Galatius (Københavns Universitet, Denmark): The action of Aut(C) on symplectic K-theory of the integers

Abstract

The symplectic group Sp_2g(Z) is the automorphism group a free abelian group Z^2g preserving a unimodular skew-symmetric form. These groups appear in topology in relation to mapping class groups of oriented 2-manifolds, and more generally from automorphisms of closed oriented manifolds of dimension 2 mod 4. They are perfect groups for g > 2, and the symplectic K-theory KSp_*(Z) may be defined as homotopy groups of the Quillen plus construction of BSp_2g(Z) in the direct limit where g goes to infinity. Work of Karoubi from the 1970s calculates these groups in terms of usual algebraic K-theory of Z, at least after completing at an odd prime p (much more recent work does this for p=2). For instance, KSp₂₂(Z) completed at p=691 is Z/691 + Z₆₉₁.
The groups Sp_2g(Z) also arise in algebraic geometry, where they are related to uniformization of principally polarized abelian varieties. This relationship may be exploited to construct a model for p-completed KSp(Z) on which the group of field automorphisms of the complex numbers naturally acts. I will discuss recent work with T. Feng and A. Venkatesh, in which we determine p-completed KSp_*(Z) as a representation of Aut(C) for odd p. In the case of KSp₂₂(Z) completed at p=691 for instance, we see that the action on Z/691 is trivial and the action on Z₆₉₁ is cyclotomic, but that the representation is a non-split extension between the two.

Back to seminar page