It is well-known that any Lie group $ G$ has $ \pi_2(G)=0$ : see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of showing it by expressing the group as an inverse limit of Lie groups?
It seems that it does not hold for all (Hausdorff) topological groups: see e.g. here (I did not go through the details…).
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