# A new formulation of the equivariant slice filtration with applications to $C_p$-slices

## TL;DR

This paper introduces a new approach to the equivariant slice filtration, providing a practical criterion for slice connectivity and explicit formulas for slices of $C_p$-spectra, advancing understanding in equivariant stable homotopy theory.

## Contribution

It offers a novel, checkable condition for slice $n$-connectivity and explicit formulas for slices of $C_p$-spectra, simplifying computations in equivariant homotopy theory.

## Key findings

- A new criterion for slice $n$-connectivity based on geometric fixed points.
- Explicit formulas for slices of $C_p$-spectra.
- Identification of minimal functors determining all slices for $C_{p^n}$-spectra.

## Abstract

This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a $G$-spectrum is slice $n$-connective. In particular, we show that a $G$-spectrum is slice greater than or equal to $n$ if and only if for all subgroups $H$, the $H$-geometric fixed points are $(n/|H|-1)$-connected. We use this to determine when smashing with a virtual representation sphere $S^V$ induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary $C_p$-spectrum and show how a very small number of functors determine all of the slices for $C_{p^n}$-spectra.