Cofreeness in Real Bordism Theory and the Segal Conjecture

TL;DR

This paper proves that certain real bordism spectra are cofree for all powers of 2, providing a new proof of the Segal conjecture for C2 using chromatic and slice techniques.

Contribution

It establishes the cofreenness of genuine C_{2^n}-spectra in real bordism and offers a novel proof of the Segal conjecture for C2 independent of Lin's theorem.

Findings

C_{2^n}-spectra are cofree for all n

New proof of the Segal conjecture for C2

Uses chromatic hypercubes and Slice Theorem techniques

Abstract

We prove that the genuine $C_{2^n}$-spectrum $N_{C_{2}}^{C_{2^n}}MU_{\mathbb{R}}$ is cofree, for all $n$. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal conjecture for $C_2$, independent of Lin's theorem.