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Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found herehere. A nice observation made there is that one can use the same arguments to show that $\pi_3(G)$ is torsion-free! (which I guess was not in your list of facts about homotopy of compact Lie groups).

Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found here. A nice observation made there is that one can use the same arguments to show that $\pi_3(G)$ is torsion-free! (which I guess was not in your list of facts about homotopy of compact Lie groups).

Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found here. A nice observation made there is that one can use the same arguments to show that $\pi_3(G)$ is torsion-free! (which I guess was not in your list of facts about homotopy of compact Lie groups).

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Renato G. Bettiol
Renato G. Bettiol
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Regarding Q3, one more explanation of why $\pi_2(G)=0$ (which is similar to what Claudio Gorodski points out above) can be found here. A nice observation made there is that one can use the same arguments to show that $\pi_3(G)$ is torsion-free! (which I guess was not in your list of facts about homotopy of compact Lie groups).