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

2020/08/11 by Lennart Meier, Meier, Lennart, Xiaolin Shi +3 · 4 citations
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2008.04963

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 C₂-Segal conjecture. We compute the localized slice spectral sequence of the C₄-norm of BP_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 N₁² H\mathbbF₂.

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• Periodicity and finite complexity in higher real K-theories
• Slice spectral sequences through synthetic spectra
• Topological Jacobi Forms
• The RO(K)-graded homotopy of Klein-four normed Mackey functors

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