2
$\begingroup$

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?

$\endgroup$

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.