On the geometric fixed-points of real topological cyclic homology

TL;DR

This paper derives a formula for the geometric fixed-points spectrum of real topological cyclic homology and applies it to key examples, confirming conjectures and aligning with known L-theory spectra.

Contribution

It provides a new formula for the geometric fixed-points spectrum of real topological cyclic homology and performs explicit computations in fundamental cases.

Findings

Calculations match known L-theory spectra where available.

The formula applies to spherical group-rings and perfect algebras.

Results support Nikolaus's conjecture on the spectrum.

Abstract

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect $\mathbb{F}_p$-algebras, and $2$-torsion free rings with perfect modulo $2$ reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.