Applications of topological cyclic homology to algebraic K-theory

TL;DR

This paper reviews the historical development and recent advances in applying topological cyclic homology to algebraic K-theory, highlighting its significance in calculations and structural understanding.

Contribution

It provides a comprehensive overview of how topological cyclic homology has evolved and contributed to algebraic K-theory from the 1970s to present, including recent breakthroughs.

Findings

Resurgence of structural theorems in K-theory

Cyclic homology variants offer insights beyond K-theory

Enhanced computational techniques for algebraic K-theory

Abstract

Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two main inroads: motivic and cyclic homology. I've been asked to present an overview of the applications of topological cyclic homology to algebraic K-theory "from a historical perspective". The timeline spans from the very early days of algebraic K-theory to the present, starting with ideas in the seventies around the "tangent space" of algebraic K-theory all the way to the current state of affair where we see a resurgence in structural theorems, calculations and a realization that variants of cyclic homology have important things to say beyond the moorings to K-theory. Comments, especially with respect to historical accuracy or missing recent contributions, are very welcome