The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture

TL;DR

This paper introduces the localized slice spectral sequence for equivariant spectra, proves its convergence, and applies it to analyze norms of Real bordism, connecting to the $C_2$-Segal conjecture and Tate differentials.

Contribution

It develops the localized slice spectral sequence, establishes its convergence, and applies it to relate Real bordism norms to the Segal conjecture, with explicit computations.

Findings

Convergence and recovery theorems for the localized slice spectral sequence.

Relation between Real bordism spectrum norms and the $C_2$-Segal conjecture.

Explicit computation of the spectral sequence for the $C_4$-norm of $BP_ eal$.

Abstract

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differentials is in one-to-one correspondence with a family of Tate differentials for $N_1^2 H{\mathbb{F}}_2$.