On the slice spectral sequence for quotients of norms of Real bordism

TL;DR

This paper develops new methods for computing equivariant homotopy groups of quotients of Real bordism spectra, with detailed calculations for specific theories related to higher real K-theories and Morava K-theories.

Contribution

It introduces novel techniques for computing equivariant slice spectral sequences of quotients of Real bordism spectra, including explicit calculations for certain $BP$-based theories.

Findings

Complete computation of the $a_{\sigma}$-localized slice spectral sequence for $BP^{((C_{2^n}))} angle m,m angle$

Establishment of a correspondence between slice spectral sequences and Adams spectral sequences

Full computation of the $a_{\lambda}$-localized slice spectral sequence for $BP^{((C_{4}))} angle 2,2 angle$

Abstract

In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm $MU^{((C_{2^n}))}$ by permutation summands. These quotients are of interest because of their close relationship with higher real $K$-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories $BP^{((C_{2^n}))}\langle m,m\rangle$. These spectra serve as natural equivariant generalizations of connective integral Morava $K$-theories. We provide a complete computation of the $a_{\sigma}$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$, where $\sigma$ is the real sign representation of $C_{2^{n-1}}$. To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. Furthermore, we provide a full computation of the $a_{\lambda}$-localized slice spectral sequence of the height-4 theory $BP^{((C_{4}))}\langle 2,2\rangle$. The $C_4$-slice spectral sequence can be entirely recovered from this computation.