The $\mathbb{Z}$-homotopy fixed points of $C_n$ spectra with applications to norms of $M{U}_{\mathbb{R}}$

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.

19 pages, 1 figure, comments welcome!