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

TL;DR

This paper develops a new computational approach to describe $Z$-homotopy fixed points of $C_n$-spectra, enabling easier calculations of their homotopy groups, with applications to norms of $MU_{R}$ and related spectra.

Contribution

Introduces a tractable method to compute $Z$-homotopy fixed points of $C_n$-spectra, extending to $N_{ abla}$-ring spectra and connecting slice spectral sequences for various spectra.

Findings

Provides a genuine $C_n$-spectrum $E^{hnZ}$ with fixed and homotopy fixed points equal.

Develops a functorial framework from divisor poset of $n$ to genuine $C_n$-spectra.

Enables explicit computation of homotopy groups for Real Johnson--Wilson theories and norms of Real bordism.

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 $N_{\infty}$-ring spectrum, this functor lifts to a functor of $N_{\infty}$-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.