Spoke topological Hochschild homology

TL;DR

This paper computes the $C_p$-equivariant spoke topological Hochschild homology of a field and demonstrates B"okstedt periodicity, providing a new proof of the Segal conjecture for cyclic groups of odd prime order.

Contribution

It introduces and computes the $C_p$-equivariant spoke topological Hochschild homology, revealing B"okstedt periodicity and offering a novel proof of the Segal conjecture for odd primes.

Findings

Confirmed B"okstedt periodicity in the spoke topological Hochschild homology.

Provided a new proof of the Segal conjecture for cyclic groups of odd prime order.

Established computational techniques for equivariant THH variants.

Abstract

Fix primes $p$ and $\ell$, and let $C_p$ be the cyclic group of order $p$. We compute the $C_p$-equivariant spoke topological Hochschild homology of $\underline{\mathbb{F}}_{\ell}$ and prove it exhibits a form of B\"okstedt periodicity. Here spoke topological Hochschild homology is a variant of topological Hochschild homology where one replaces the circle in the construction with the unreduced suspension of $C_p$. As an application, we use this result to give a new proof of the Segal conjecture for the cyclic group of order an odd prime $p$.