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

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 $a_σ$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$, where $σ$ 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 $a_λ$-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.

Figures (15)

• Figure 1: The $a_\sigma$-localized slice spectral sequence of $a_{\sigma}^{-1}BP_{{\mathbb R}}\langle 2\rangle$ in integer degrees. The slice spectral sequence of $BP_{{\mathbb R}}\langle 2\rangle$ is obtained by removing the region below the horizontal line $s=0$ and replacing $\bullet={\mathbb Z}/2$ by copies of $\mathbb{Z}$, which reintroduces the transfers.
• Figure 2: The $a_{\sigma}$-localized slice spectral sequences of $a_{\sigma}^{-1}BP_{{\mathbb R}}\langle 2,2\rangle$ (top). The middle figure is the slice spectral sequence of $BP_{{\mathbb R}}\langle 2,2\rangle$ and the bottom is its $E_{\infty}$-page. A $\square$ denotes ${\mathbb Z}_{(2)}$, a $\bullet$ denotes ${\mathbb Z}/2$.
• Figure 3: The $a_\sigma$-localized slice spectral sequence of $a_{\sigma}^{-1}BP^{(\!(C_4)\!)}\langle 2 \rangle$.
• Figure 4: The $a_\sigma$-localized slice spectral sequence of $a_{\sigma}^{-1}BP^{(\!(C_4)\!)}\langle 2,2 \rangle$.
• Figure 5: The $a_\sigma$-localized slice spectral sequence of $a_\sigma^{-1}i_{C_2}^*BP^{(\!(C_4)\!)}\langle 2,2 \rangle$.

Theorems & Definitions (125)

• Theorem A: Theorem \ref{['prop:SlicesOfRegularQuotients']}
• Theorem B: Theorem \ref{['thm:BPGmmHeight']}
• Theorem C: Theorem \ref{['prop:geofixall']} and Theorem \ref{['thm:unlocalizingasigma']}
• Theorem D: Theorem \ref{['prop:notaring']}
• Theorem E: Theorem \ref{['thm:adamsres']}, Corollary \ref{['cor:speedupIsom']}, and Corollary \ref{['cor:redindex']}
• Theorem F: Theorem \ref{['thm:diffs']} and Summary \ref{['sum:relvsslice']}
• Theorem G
• Conjecture 1.7
• Definition 2.1
• Definition 2.2