# \'Etale homotopy groups of algebraic groups and homogeneous spaces

## TL;DR

This paper proves the vanishing of the second étale homotopy group for smooth algebraic groups over separably closed fields and derives an explicit formula for homogeneous spaces, extending classical topological results to algebraic geometry.

## Contribution

It establishes a new algebraic analogue of Cartan's theorem and provides explicit calculations for the second étale homotopy group of homogeneous spaces.

## Key findings

- Vanishing of the second étale homotopy group for algebraic groups.
- Explicit formula for the second étale homotopy group of homogeneous spaces.
- Extension of classical topological theorems to algebraic geometry.

## Abstract

We show the vanishing of the second homotopy group of the \'etale 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.