The \mathbb Z-homotopy fixed points of C_n spectra with applications to norms of MU_{\mathbb R}

2018/08/30 by Michael A. Hill, Hill, Michael A., Mingcong Zeng +1
Mathematics · Medicine · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.1808.10412

openalex publication_date 2018/08/30 · openalex created_date 2018/09/07 · openalex updated_date 2026/07/28

Abstract

We introduce a computationally tractable way to describe the \mathbb Z-homotopy fixed points of a C_n-spectrum E, producing a genuine C_n spectrum E^{hn\mathbb Z} whose fixed and homotopy fixed points agree and are the \mathbb Z-homotopy fixed points of E. These form a piece of a contravariant functor from the divisor poset of n to genuine C_n-spectra, and when E is an N_∞-ring spectrum, this functor lifts to a functor of N_∞-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the RO(G)-graded homotopy groups of the spectrum E^{hn\mathbb Z}, giving the homotopy groups of the \mathbb Z-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple \mathbb Z-homotopy fixed point case, giving us a family of new tools to simplify slice computations.

Citations

• On the nonexistence of elements of Kervaire invariant one

Related

• Slice spectral sequences through synthetic spectra
• The slice spectral sequence for the C₄ analog of real K-theory
• The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture
• The homotopy fixed point spectra of profinite Galois extensions
• Several homotopy fixed point spectral sequences in telescopically localized algebraic K-theory