On A^1-fundamental groups of isotropic reductive groups
Konrad Voelkel, Matthias Wendt

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 -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 -fundamental group. Using -homotopy theory, we deduce a Steinberg relation for these explicit loops.
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