A stratification of the equivariant slice filtration

Abstract

In this paper, we construct a stratification tower for the equivariant slice filtration. This tower stratifies the slice spectral sequence of a $G$-spectrum $X$ into distinct regions. Within each of these regions, the differentials are determined by the localized slice spectral sequences, which compute the geometric fixed points along with their associated residue group actions. Consequently, the stratification tower offers an inductive method of understanding the entirety of the equivariant slice spectral sequence of $X$ by examining each of its distinct stratification regions.

Figures (10)

• Figure 1: Hill--Hopkins--Ravenel's computation of the $C_4$-slice spectral sequence of $E_2^{hC_4}$.
• Figure 2: Stratification regions of the slice spectral sequence.
• Figure 3: The isomorphism region $\mathcal{R}_{\mathcal{F}, \mathcal{F}'}$ for the map $\varphi$.
• Figure 4: Stratification regions of the slice spectral sequence when $V = 0$.
• Figure 5: The stratification tower of the slice spectral sequence.

Theorems & Definitions (31)

• Theorem A: Stratification
• Theorem B: Comparison, \ref{['thm:ROGSliceIsom2']}
• Theorem C: Slice Recovery Theorem, \ref{['thm:SliceRecovery1']}
• Theorem D: Positive Cone, \ref{['thm:SliceSSVanishingRegion']}
• Proposition 2.1
• proof
• Definition 2.2
• Corollary 2.3
• Remark 2.4
• Definition 2.5