# The $p$-completed cyclotomic trace in degree $2$

## TL;DR

This paper proves that for certain algebraic structures, the cyclotomic trace map on the second homotopy group acts as a q-deformed logarithm, revealing a deep connection between algebraic K-theory and topological cyclic homology.

## Contribution

It establishes a precise identification of the cyclotomic trace map on rac{2}{ ext{pi}}_2 with a q-deformation of the logarithm for quasi-regular semiperfectoid algebras.

## Key findings

- The cyclotomic trace map on rac{2}{ ext{pi}}_2 is a q-deformation of the logarithm.
- The result applies to quasi-regular semiperfectoid rac{ ext{Z}_p^{ ext{cycl}}}{ ext{algebra} } R.
- Provides a new perspective on the relationship between algebraic K-theory and topological cyclic homology.

## Abstract

We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological cyclic homology $\mathrm{TC}(R;\mathbb{Z}_p)$ of $R$ identifies on $\pi_2$ with a $q$-deformation of the logarithm.