The fundamental group of semisimple group orbits

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It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such that $G/H$ is compact where $H$ is a closed subgroup. Then is it true that $pi_1(G/H)$ is finite?




lie-groups topological-groups




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    It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such that $G/H$ is compact where $H$ is a closed subgroup. Then is it true that $pi_1(G/H)$ is finite?




    lie-groups topological-groups




    share|cite|improve this question





















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      up vote
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      down vote

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      It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such that $G/H$ is compact where $H$ is a closed subgroup. Then is it true that $pi_1(G/H)$ is finite?




      lie-groups topological-groups




      share|cite|improve this question











      It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such that $G/H$ is compact where $H$ is a closed subgroup. Then is it true that $pi_1(G/H)$ is finite?





      lie-groups topological-groups





      share|cite|improve this question










      share|cite|improve this question




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      Ronald

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