Étale homotopy groups of algebraic groups and homogeneous spaces

Abstract

We show the vanishing of the second homotopy group of the étale homotopy type of a smooth connected algebraic group over a separably closed field, completed away from the characteristic. This is an algebraic analogue of a classical theorem of Elie Cartan. Based on this result, we establish an explicit formula for the similarly completed second homotopy group of a homogeneous space.

arXiv commentv2: major revision, some inaccurate statements in the first version corrected

References in corpus (2)

• Une formule pour le groupe de Brauer algébrique d'un torseur
• Le groupe fondamental d'un espace homogène d'un groupe algébrique linéaire