The Equivariant Slice Filtration: a Primer

TL;DR

This paper introduces the equivariant slice filtration, explores its properties, and applies it to spectra related to Mackey functors, establishing connections with Postnikov towers and proposing new conjectures.

Contribution

It provides a comprehensive introduction to the equivariant slice filtration, including new results on slice dimensions, pullbacks, and the slice tower for specific spectra.

Findings

Determined slice dimensions for various spectra

Analyzed pullbacks of slices on quotient groups

Established the slice tower for Eilenberg-Mac Lane spectra

Abstract

We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice dimension of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg-Mac Lane spectrum associated to a Mackey functor for a cyclic $p$-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks.