Étale homotopy groups of algebraic groups and homogeneous spaces

2017/01/17 by Cyril Demarche, Demarche, Cyril, Tamás Szamuely +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1701.04867

openalex publication_date 2017/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Le groupe fondamental d'un espace homogène d'un groupe algébrique linéaire

Related

• The first and second homotopy groups of a homogeneous space of a complex linear algebraic group
• On schemes with trivial higher étale homotopy groups
• Homogeneous spaces over an abelian variety
• Prime-to-p étale covers of algebraic groups
• On the homotopy groups of p-completed classifying spaces