### Topological cyclic homology

This handbook chapter gives an introduction to topological cyclic
homology following the modern setup of Nikolaus-Scholze. It contains a
proof of the Bökstedt periodicity theorem close to his original
unpublished proof, as well as the proof by Bhatt-Morrow-Scholze of its
extension to all perfectoid rings. The latter is used to give a
purely *p*-adic proof of the Bott periodicity theorem. Finally,
the cofiber of the assembly map in *p*-adic *K*-theory
for a cyclic group of order *p* and a perfectoid ring of
coefficients is calculated.

Lars Hesselholt
<larsh@math.nagoya-u.ac.jp>