Topology seminar

Speaker: Filipp Levikov (Universität Münster)

Title: Quadratic L-theory and the fibre of the cyclotomic trace

Abstract:

The cyclotomic trace map, as constructed by Bockstedt, Hsiang and Madsen is a natural transformation from algebraic K-theory to topological cyclic homology TC with a locally constant fibre. While the original construction uses a signifi- cant amount of equivariant stable homotopy theory an alternative model for com- pleted TC of rings and the corresponding trace map was described by Betley and Schlichtkrull.

Using this one can construct a model for the completed trace map which respects the canonical involution on K-theory. Furthermore the model of Betley-Schlichtkrull suggests a way to connect the fibre of the cyclotomic trace to quadratic L-theory. Time permitting, I will explain how a theorem of Grayson on K-theory of endomor- phisms comes into play and how the construction of the non-linear projective line introduced by Hu ̈ttemann and coauthors has to be adjusted to prove a non-linear analogue of Grayson’s theorem which also takes involutions into account.