It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory?
Also, who gave the first proof of this theorem?
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This was discussed on MathOverflow a while ago. Have you looked there? – Mariano Suárez-Álvarez Aug 19 '13 at 16:15
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2Jason DeVito gives a lovely proof using Morse theory at this MO post. The most homotopy theory used is the Hurewicz Theorem and the fibration of a pathspace with loopspace fibers. – Dan Rust Aug 19 '13 at 19:45