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

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