On the Slice Spectral Sequence

TL;DR

This paper introduces a variant of the slice spectral sequence focusing on regular slice cells, explores its relationship with the original, and applies it to analyze equivariant spectra and prove a conjecture about Eilenberg MacLane spectra.

Contribution

It develops a new version of the slice spectral sequence using only regular slice cells and establishes its relationship with the classical sequence, advancing equivariant stable homotopy theory.

Findings

The regular slice spectral sequence is related to the classical one.

The slice filtration of spectra over a normal subgroup relates to geometric fixed points.

Proved Hill's conjecture on the slice filtration of Eilenberg MacLane spectra.

Abstract

We introduce a variant of the slice spectral sequence which uses only regular slice cells, and state the precise relationship between the two spectral sequences. We analyze how the slice filtration of an equivariant spectrum that is concentrated over a normal subgroup is related to the slice filtration of its geometric fixed points, and use this to prove a conjecture of Hill on the slice filtration of an Eilenberg MacLane spectrum. We also show how the (co)connectivity of a spectrum results in the (co)connectivity of its slice tower, demonstrating the "efficiency" of the slice spectral sequence.