# Topological cyclic homology

## TL;DR

This survey introduces topological cyclic homology, discusses key theorems like Bökstedt periodicity, and explores recent extensions and applications in p-adic contexts, providing foundational and advanced insights into the subject.

## Contribution

It offers a comprehensive overview of topological cyclic homology, including proofs of fundamental theorems and recent extensions to perfectoid rings, with applications in p-adic homotopy theory.

## Key findings

- Proof of Bökstedt periodicity resembling original proof
- Extension of Bökstedt periodicity to perfectoid rings
- Evaluation of the cofiber of the assembly map in p-adic TC

## Abstract

This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\"okstedt periodicity that closely resembles B\"okstedt's original unpublished proof. We explain the extension of B\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.