The Structure of Motivic Homotopy Groups

TL;DR

This paper investigates the structure of stable motivic homotopy groups over complex numbers, revealing their division into four distinct regions with varying degrees of understanding, including known, classical, and unknown parts.

Contribution

It provides a detailed analysis of the motivic homotopy groups' structure over , identifying regions with different levels of knowledge and relating them to classical homotopy groups.

Findings

The motivic homotopy groups over  are divided into four regions.

The ta-local region is completely understood.

The ta-local region matches classical stable homotopy groups.

Abstract

We study the stable motivic homotopy groups $\pi_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an $\eta$-local region that is entirely known, a $\tau$-local region that is identical to classical stable homotopy groups, and a region that is not well-understood.