On categories of slices

TL;DR

This paper provides an algebraic framework for understanding the category of $n$-slices in equivariant stable homotopy theory, introducing explicit constructions and revealing new algebraic properties of associated Green functors.

Contribution

It constructs an explicit $G$-spectrum called isotropic slice $n$-sphere and characterizes $n$-slices as modules over a Green functor, simplifying their algebraic description.

Findings

Category of $n$-slices is equivalent to certain modules over a Green functor.

Introduces isotropic slice $n$-sphere with explicit properties.

Green functors for certain slices are 'geometrically split', reducing data complexity.

Abstract

In this paper we give an algebraic description of the category of $n$-slices for an arbitrary group $G$, in the sense of Hill-Hopkins-Ravenel. Specifically, given a finite group $G$ and an integer $n$, we construct an explicit $G$-spectrum $W$ (called an isotropic slice $n$-sphere) with the following properties: (i) the $n$-slice of a $G$-spectrum $X$ is equivalent to the data of a certain quotient of the Mackey functor $\underline{[W,X]}$ as a module over the endomorphism Green functor $\underline{[W,W]}$; (ii) the category of $n$-slices is equivalent to the full subcategory of right modules over $\underline{[W,W]}$ for which certain restriction maps are injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors $\underline{[W,W]}$ for certain slice $n$-spheres have a special property (they are "geometrically split") which reduces the amount of data necessary to specify a $\underline{[W,W]}$-module. This step is purely algebraic and may be of independent interest.