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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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