The first and second homotopy groups of a homogeneous space of a complex linear algebraic group

TL;DR

This paper computes the first and second topological homotopy groups of homogeneous spaces of complex linear algebraic groups, providing explicit descriptions when the stabilizer subgroup is connected.

Contribution

It offers explicit calculations of the fundamental and second homotopy groups for a broad class of homogeneous spaces of complex algebraic groups, extending previous understanding.

Findings

Computed the topological fundamental group for homogeneous spaces with connected stabilizers.

Determined the second homotopy group for these spaces.

Extended known results to a wider class of algebraic group actions.

Abstract

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we compute the topological fundamental group $\pi_1^{\rm top}(X({\mathbb C}),x)$. Moreover, we compute the second homotopy group $\pi_2^{\rm top}(X({\mathbb C}),x)$.