On the zero slice of the sphere spectrum

TL;DR

This paper proves that over fields of characteristic zero, the zero slice of the motivic sphere spectrum is the motivic Eilenberg-MacLane spectrum, with implications for the structure of slices of any spectrum.

Contribution

It establishes the identification of the zero slice of the motivic sphere spectrum with the motivic Eilenberg-MacLane spectrum in characteristic zero.

Findings

Zero slice of motivic sphere spectrum equals motivic Eilenberg-MacLane spectrum

Slices of any spectrum are modules over the motivic Eilenberg-MacLane spectrum

Analysis of symmetric powers of the T-sphere underpins the proof

Abstract

In this paper we prove over fields of characteristic zero that the zero slice of the motivic sphere spectrum is the motivic Eilenberg-Maclane spectrum. As a corollary one concludes that the slices of any spectrum are modules over the motivic Eilenberg-MacLane spectrum. To prove our result we analyze the unstable homotopy type of the symmetric powers of the T-sphere.