On A^1-fundamental groups of isotropic reductive groups

TL;DR

This paper studies the $A^1$-fundamental groups of isotropic reductive groups over fields, linking them to second homology groups and providing explicit generators and relations.

Contribution

It establishes a connection between $A^1$-fundamental groups and second homology for isotropic reductive groups, and constructs explicit loops for split groups.

Findings

Sections of $A^1$-fundamental sheaf correspond to second homology groups

Explicit loops are constructed for split groups

Steinberg relations are derived for these loops

Abstract

For an isotropic reductive group G satisfying a suitable rank condition over an infinite field k, we show that the sections of the $\mathbb{A}^1$-fundamental group sheaf of G over an extension field L/k can be identified with the second group homology of G(L). For a split group G, we provide explicit loops representing all elements in the $\mathbb{A}^1$-fundamental group. Using $\mathbb{A}^1$-homotopy theory, we deduce a Steinberg relation for these explicit loops.