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...).
EDIT: according to a deep result by Malcev-Iwasawa, any connected locally compact group deformation retracts to a compact subgroup, so the locally compact case reduces to the compact one.