On A^1-fundamental groups of isotropic reductive groups

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.

shortened to 5 pages, using excision/descent theory from Asok-Hoyois-Wendt

References in corpus (1)

• Homotopy invariance of non-stable K_1-functors

Cited by in corpus (6)

• Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces
• Homotopy invariance of non-stable K_1-functors
• On the ${\mathbb{A}}^1$-invariance of ${\mathrm{K}}_2$ modeled on linear and even orthogonal groups
• R-equivalence and A^1-connectedness in anisotropic groups
• On the ${\mathbb{A}}^1$-invariance of ${\mathrm{K}}_2$ of Chevalley groups of simply-laced type
• On third homology of SL_2 and weak homotopy invariance