Stable motivic pi_1 of low-dimensional fields

TL;DR

This paper computes the first stable motivic homotopy group of spheres over low-dimensional fields, confirming Morel’s pi_1-conjecture in these cases and expanding understanding of motivic pi_1 in various weights.

Contribution

It determines the 1-column of the motivic Adams-Novikov spectral sequence over low-dimensional fields and verifies Morel’s pi_1-conjecture for these fields.

Findings

Confirmed Morel's pi_1-conjecture for low-dimensional fields

Computed the first stable motivic homotopy group of spheres over such fields

Determined stable motivic pi_1 in multiple weights

Abstract

Let k be a field with cohomological dimension less than 3; we call such fields low-dimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1-column of the motivic Adams-Novikov spectral sequence over k. Combined with rational information we use this to compute the first stable motivic homotopy group of the sphere spectrum over k. Our main result affirms Morel's pi_1-conjecture in the case of low-dimensional fields. We also determine stable motivic pi_1 in integer weights other than -2, -3, and -4.