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Journal of Lie Theory 33 (2023), No. 2, 687--700
Copyright Heldermann Verlag 2023

The First and Second Homotopy Groups of a Homogeneous Space of a Complex Linear Algebraic Group

Mikhail Borovoi
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Israel


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

Keywords: Fundamental group, second homotopy group, homogeneous space, linear algebraic group.

MSC: 14F35, 14M17, 20G20.

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