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On \(\mathbb{A}^1\)-fundamental groups of isotropic reductive groups. (Sur le groupe fondamental au sens de la A1-homotopie des groupes réductifs isotropes.) (English. French summary) Zbl 1387.14065
Summary: 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.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
20G15 Linear algebraic groups over arbitrary fields
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