An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of calculating $\pi_2(G)$ and $\pi_3(G)$ proceeds via the Bott and Samelson construction of a perfect Morse function on $G/T$ (Perhaps the more standard way is via Iwasawa decompositions and the description of $G/T$ as a complex manifold, but my student says he prefers "geometry" to "algebra").

Bott and Samelson's papers are of course wonderful but aside from being rather vintage, they are research papers and they discuss a lot of applications to loop spaces, whose infinite dimensional nature I think would be intimidating for this student. I am therefore looking for a reference that I can give to my student that discusses this construction in a fairly "hands-on" and friendly fashion and has relatively complete proofs up to perhaps isolating one or two facts from Lie theory (say about the $\operatorname{Ad}$ representation or the Weyl group). Any suggestions?

  • $\begingroup$ Take $G$ be connected and simply-connected. From the principal $G_a$-bundle $G_a\rightarrow G\rightarrow G/G_x=\mathcal{O}_x$, where $\mathcal{O}_x$,is the coadjoint orbit we obtain a long-exact sequence $$\ldots\rightarrow\pi_i(G_x)\rightarrow\pi_i(G)\rightarrow\pi_i(\mathcal{O}_x) \rightarrow \pi_{i-1}(G_x)\rightarrow\ldots.$$ Since $\pi_1(G)=0$ and $\pi_2(G)=0$, exactness considerations imply that $\pi_1(G_x)=\pi_2(\mathcal{O}_x)$. $\endgroup$
    – user21574
    Jan 27 '16 at 23:31
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    $\begingroup$ Maybe Sect. 3.4 of these notes www3.nd.edu/~lnicolae/Morse2nd.pdf might be more appealing to your student. $G/T$ is a coadjoint orbit and as such it is equipped with many perfect Morse functions. In these notes I also discuss special coadjoint orbits such as Grassmannians and flag manifolds. $\endgroup$ Jan 27 '16 at 23:36
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    $\begingroup$ @HassanJolany As I mentioned in the question, the goal is to \emph{prove} $\pi_2(G)=0$. There are of course other ways to do this, but one standard way is via using some suitable Bruhat or Morse stratification to a priori prove that $\pi_2(G/T)$ is torsion-free. $\endgroup$
    – user36931
    Jan 28 '16 at 0:26
  • $\begingroup$ @user36931 It is known fact from Olivier Debarre that any smooth, complex, rationally connected projective variety is simply connected. So, because $G/T$ is rationally coonected projective variety , so $\pi_1(G/T)=e$ $\endgroup$
    – user21574
    Jan 28 '16 at 0:34
  • $\begingroup$ @LiviuNicolaescu Thank you for the notes. This symplectic point of view is very nice and of course could be an entry point into other subjects. $\endgroup$
    – user36931
    Jan 28 '16 at 0:35

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