Generalized \(\mathbb{Z}\)-homotopy fixed points of \(C_n\) spectra with applications to norms of \(MU_{\mathbb{R}}\)
type: Item
instance of: scholarly article
title: Generalized \(\mathbb{Z}\)-homotopy fixed points of \(C_n\) spectra with applications to norms of \(MU_{\mathbb{R}}\)
zbMATH Open document ID: 1479.55024
author: Michael A. Hill
author: Mingcong Zeng
published in: The New York Journal of Mathematics
publication date: 2020-01-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1808.10412
full work available at URL: 26-4.html
review text: In this paper, the authors introduce a way to describe the \(\mathbb{Z}\)-homotopy fixed points of a \(C_n\)-spectrum. In more detail, let \(E\) be a \(C_n\)-spectrum. A choice of generator for the cyclic group \(C_n\) gives a surjection \(\mathbb{Z} \to C_n\), and it is therefore possible to form the \(\mathbb{Z}\)-homotopy fixed points of \(E\). The first main result of the authors is that, from this data, one can associate a cofree \(C_n\)-spectrum \(E^{hn\mathbb{Z}}\) such that for all subgroups \(C_d \subseteq C_n\) we have \[ (E^{hn\mathbb{Z}})^{C_d} \simeq E^{h(\frac{n}{d}\mathbb{Z})}. \] The \(C_n\)-spectrum \(E^{hn\mathbb{Z}}\) is the quotient of \(E\) by the Euler class associated to the faithful irreducible representation of \(C_n\) given by choosing a primitive \(n\)-th root of unity. Choosing different representations of \(C_n\) and quotienting by the associated Euler class, the authors produce other interesting spectra. The second part of the paper is dedicated to applications. For example, let \(E_{\mathbb{R}}(n)\) denote the real Johnson-Wilson theory studied by \textit{N. Kitchloo} and \textit{W. S. Wilson} [Geom. Topol. Monogr. 10, 237--244 (2007; Zbl 1117.55001)]. The authors identify the homotopy fixed points \(E_{\mathbb{R}}(n)^{h\mathbb{Z}}\) as certain quotients of \(EO(n) = E_{\mathbb{R}}(n)^{C_2}\). This should be thought of as a higher analog of the identification of self conjugate \(K\)-theory as \(KO/\eta^2\). They also study norms of the real bordism spectrum \(MU_{\mathbb{R}}\), made famous by the Hill-Hopkins-Ravenel solution of the Kervaire invariant one problem [\textit{M. A. Hill} et al., Ann. Math. (2) 184, No. 1, 1--262 (2016; Zbl 1366.55007)].
reviewed by: Drew Heard
Mathematics Subject Classification ID: 55Q91
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55T99
zbMATH DE Number: 7160439
zbMATH Keywords: equivariant homotopy
zbMATH Keywords: slice spectral sequence
zbMATH Keywords: homotopy fixed points
MaRDI profile type: Publication
arXiv ID: 1808.10412
cites work: Graded Tambara functors
cites work: K-THEORY AND REALITY
cites work: Operadic multiplications in equivariant spectra, norms, and transfers
cites work: Incomplete Tambara functors
cites work: A classification of K-local spectra
cites work: Witt vectors and equivariant ring spectra applied to cobordism
cites work: Q4101174
cites work: A cohomology theory based upon self-conjugacies of complex vector bundles
cites work: Equivariant iterated loop space theory and permutative \(G\)-categories
cites work: Encoding equivariant commutativity via operads
cites work: Real orientations of Lubin-Tate spectra
cites work: The Arf-Kervaire problem in algebraic topology: sketch of the proof
cites work: On the nonexistence of elements of Kervaire invariant one
cites work: The slice spectral sequence for the \(C_{4}\) analog of real \(K\)-theory
cites work: The \(C_2\)-spectrum \(\mathrm{Tmf}_1(3)\) and its invertible modules
cites work: Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence
cites work: On fibrations related to real spectra
cites work: The second real Johnson-Wilson theory and nonimmersions of \(RP^n\). II.
cites work: Conjugations on complex manifolds and equivariant homotopy of 𝑀𝑈
cites work: Periodic phenomena in the Adams-Novikov spectral sequence
cites work: Q3747338
cites work: Invariants and semi-direct products for finite group actions on tensor categories
Recommended article: The real Johnson-Wilson cohomology of \(\mathbb{CP}^\infty\)
Recommended article: Multiplicative structure on Real Johnson-Wilson theory
Recommended article: TheER(n)-cohomology ofBO(q) and real Johnson–Wilson orientations for vector bundles
Recommended article: On the Hopf ring for \(ER(n)\)
Recommended article: On fibrations related to real spectra
Recommended article: The ER(z)-cohomology of Bℤ/(2q) and ℂℙn
Recommended article: Landweber flat real pairs and \(ER(n)\)-cohomology
Recommended article: On real-oriented Johnson-Wilson cohomology
Recommended article: Formal completion of the Jacobians of plane curves and higher real \(K\)-theories
Recommended article: The \textit{ER}(2)-cohomology of \(\prod^n \mathbb{CP}^\infty\) and \(BU (n)\)
description: scientific article