Let $G$ be a (connected, WLOG compact) lie group. I have seen it stated (without proof) that all of it's homotopy groups are finitely generated. I have no clue how to prove this even for $\pi_1$. Once this is settled the general case can probably be done by assuming that $G$ is simply connected and compact (by passing to the universal cover and then to a maximal compact subgroup) and arguing more generally that every compact simply connected (smooth) manifold has finitely generated homotopy groups, though I do not know how to prove such a statement. I have a feeling that one can use the fact that the homologies are f.g and some sort of Hurewicz (and maybe some spectral sequence argument) to show this, but I don't really understand how. So my questions are basically:

  1. How does one prove that a Lie group has finitely generated fundamental group? and:
  2. How to show that a simply connected compact (smooth) manifold has finitely generated homotopy groups?

Any reference/explicit argument will be appreciated. $$$$ [edit] Of course the fundamental group of $G$, being abelian, must be isomorphic to the first homology which is finitely generated. Hence my first question is completely stupid. what about the second?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.