Real topological Hochschild homology and the Segal conjecture

TL;DR

This paper provides a new proof of the Segal conjecture for the cyclic group of order two using calculations of Real topological Hochschild homology, avoiding reliance on Lin's theorem.

Contribution

It introduces a novel approach based on Real topological Hochschild homology calculations to prove the Segal conjecture independently of Lin's theorem.

Findings

Calculated the Real topological Hochschild homology of F_2 as a Hopf algebroid.

Determined the E_2-page of the descent spectral sequence for the norm map.

Established a new upper bound on the RO(C_2)-graded homotopy of the norm of F_2.

Abstract

We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of $\mathbb{F}_2$. This determines the $\mathrm{E}_2$-page of the descent spectral sequence for the map $\mathrm{N}\mathbb{F}_2 \to \mathbb{F}_2$, where $\mathrm{N}\mathbb{F}_2$ is the $C_2$-equivariant Hill--Hopkins--Ravenel norm of $\mathbb{F}_2$. The $\mathrm{E}_2$-page represents a new upper bound on the $RO(C_2)$-graded homotopy of $\mathrm{N}\mathbb{F}_2$, from which the Segal conjecture is an immediate corollary.