Higher topological cyclic homology and the Segal conjecture for tori

TL;DR

This paper develops higher topological cyclic homology based on n-dimensional tori, introduces new operators, and uses these tools to compute key invariants, ultimately proving the Segal conjecture for tori.

Contribution

It introduces a higher analog of topological cyclic homology with new operators and relations, and applies these to establish the Segal conjecture for the torus.

Findings

Computed topological restriction and Frobenius homology for the sphere spectrum.

Established the Segal conjecture for the torus.

Analyzed relations among restriction, Frobenius, Verschiebung, and differentials.

Abstract

We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory. The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology. We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.