The topological cyclic homology of the dual circle

TL;DR

This paper provides a new proof of a duality result for the circle in spectra and computes its topological cyclic homology, impacting conjectures in algebraic K-theory.

Contribution

It offers a novel proof of Lazarev's duality result and calculates the topological cyclic homology of the dual circle spectrum.

Findings

Dual of the circle spectrum is a strictly square-zero extension

Calculated topological cyclic homology of the dual circle spectrum

Ruled out a Koszul-dual reformulation of the Novikov conjecture

Abstract

We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.