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.