Homotopy groups of Lie groups

Question

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, perhaps like a more clever version of the proof that $\pi_1(G)$ must be abelian?

Answer 1

I don't know of anything as bare hands as the proof that $\pi_1(G)$ must be abelian, but here's a sketch proof I know (which can be found in Milnor's Morse Theory book. Plus, as an added bonus, one learns that $\pi_3(G)$ has no torsion!):

First, (big theorem): Every (connected) Lie group deformation retracts onto its maximal compact subgroup (which is, I believe, unique up to conjugacy). Hence, we may as well focus on compact Lie groups.

Let $PG = \{ f:[0,1]\rightarrow G | f(0) = e\}$ (I'm assuming everything is continuous.). Note that $PG$ is contractible (the picture is that of sucking spaghetti into one's mouth). The projection map $\pi:PG\rightarrow G$ given by $\pi(f) = f(1)$ has homotopy inverse $\Omega G = $Loop space of G = $\{f\in PG | f(1) = e \}$.

Thus, one gets a fibration $\Omega G\rightarrow PG\rightarrow G$ with $PG$ contractible. From the long exact sequence of homotopy groups associated to a fibration, it follows that $\pi_k(G) = \pi_{k-1}\Omega G$

Hence, we need only show that $\pi_{1}(\Omega G)$ is trivial. This is where the Morse theory comes in. Equip $G$ with a biinvariant metric (which exists since $G$ is compact). Then, following Milnor, we can approximate the space $\Omega G$ by a nice (open) subset $S$ of $G\times\cdots \times G$ by approximating paths by broken geodesics. Short enough geodesics are uniquely defined by their end points, so the ends points of the broken geodesics correspond to the points in $S$. It is a fact that computing low (all?...I forget)* $\pi_k(\Omega G)$ is the same as computing those of $S$.

Now, consider the energy functional $E$ on $S$ defined by integrating $|\gamma|^2$ along the entire curve $\gamma$. This is a Morse function and the critical points are precisely the geodesics**. The index of E at a geodesic $\gamma$ is, by the Morse Index Lemma, the same as the index of $\gamma$ as a geodesic in $G$. Now, the kicker is that geodesics on a Lie group are very easy to work with - it's pretty straight forward to show that the conjugate points of any geodesic have even index.

But this implies that the index at all critical points is even. And now THIS implies that $S$ has the homotopy type of a CW complex with only even cells involved. It follows immediately that $\pi_1(S) = 0$ and that $H_2(S)$ is free ($H_2(S) = \mathbb{Z}^t$ for some $t$).

Quoting the Hurewicz theorem, this implies $\pi_2(S)$ is $\mathbb{Z}^t$.

By the above comments, this gives us both $\pi_1(\Omega G) = 0$ and $\pi_2(\Omega G) = \mathbb{Z}^t$, from which it follows that $\pi_2(G) = 0$ and $\pi_3(G) = \mathbb{Z}^t$.

Incidentally, the number $t$ can be computed as follows. The universal cover $\tilde{G}$ of $G$ is a Lie group in a natural way. It is isomorphic (as a manifold) to a product $H\times \mathbb{R}^n$ where $H$ is a compact simply connected group.

H splits isomorphically as a product into pieces (all of which have been classified). The number of such pieces is $t$.

(edits)

*- it's only the low ones, not "all", but one can take better and better approximations to get as many "low" k as one wishes.

**- I mean CLOSED geodesics here

Answer 2

The elementary proof that $\pi_1$ is abelian applies more generally to H-spaces (spaces $X$ with a continuous multiplication map $X \times X \to X$ having a 2-sided identity element) without any assumption of finite dimensionality, but infinite-dimensional H-spaces can have nontrivial $\pi_2$, for example $CP^\infty$ (which can be replaced by a homotopy equivalent topological group if one wants, as Milnor showed). Thus finite-dimensionality is essential, so any proof would have to be significantly less elementary than for the $\pi_1$ statement. It is a rather deep theorem of W.Browder (in the 1961 Annals) that $\pi_2$ of a finite-dimensional H-space is trivial.

Hopf's theorem that a finite-dimensional H-space (with finitely-generated homology groups) has the rational homology of a product of odd-dimensional spheres implies that $\pi_2$ is finite, but the argument doesn't work for mod p homology so one can't rule out torsion in $\pi_2$ so easily. It's not true that a simply-connected Lie group is homotopy equivalent to a product of odd-dimensional spheres. For example the mod 2 cohomology ring of Spin(n) is not an exterior algebra when n is sufficiently large. For SU(n) the cohomology ring isn't enough to distinguish it from a product of spheres, but if SU(n) were homotopy equivalent to a product of odd-dimensional spheres this would imply that all odd-dimensional spheres were H-spaces (since a retract of an H-space is an H-space) but this is not true by the Hopf invariant one theorem. There are probably more elementary arguments for this.

Answer 3

Here's another proof based on the structure of the flag variety $G/T$ of $G$. A compact Lie group $G$ has a maximal torus $T$, and $G$ is a principal $T$-bundle over the quotient $G/T$. Borel showed that $G/T$ is a complex manifold, and gave a CW decomposition of it with no odd-dimensional cells. (This is not deep but still astonishing, and the start of a long story; I like the context given by Hirzebruch's eulogy for Borel, available on page 9 here.)

Since $\pi_2(T) = 0$, we have an exact sequence

$$0 \to \pi_2(G) \to \pi_2(G/T) \to \pi_1(T)$$

We can conclude immediately that $\pi_2(G)$ is torsion-free, since $\pi_2(G/T) = H_2(G/T)$ is a free group on the 2-cells in $G/T$. After Allen's answer (Hopf's theorem) this shows $\pi_2(G) = 0$.

With a little more Lie theory one can show directly that the connecting map $\pi_2(G/T) \to \pi_1(T)$ is injective. The group $\pi_1(T)$ has a linearly independent subset of simple coroots, and the 2-cells in $G/T$ are indexed by simple roots. The connecting homomorphism matches these up in the natural way, which one can see by considering rank 1 subgroups (subgroups of the form $SU(2)$ or $PSU(2)$) of $G$. As a consequence you get a formula for $\pi_1(G)$ in terms of roots and coroots.

Answer 4

For the classical Lie groups, I think that an easy way to obtain the result is through the fibrations:

$SO(n-1)\to SO(n)\to S^{n-1}$,

$SU(n-1)\to SU(n)\to S^{2n-1}$,

$SP(n-1)\to SP(n)\to S^{4n-1}$

and the homotopy long exact sequence and $\pi_m(S^n) = 0$ for $m$ less than $n$, and $\pi_2(SO(2)) = \pi_2(SU(2))=0$ and the isomorphism of $SP(2)$ and $SO(5)$.

Answer 5

There's a proof that $\pi_2(G)$ is trivial for compact semi-simple Lie groups in section 8.6 of Pressley and Segal's Loop Groups. They say "This proof is in essence the same as Bott's Morse theory proof" but it has some differences in treatment even if the substance is the same. The major difference is that they don't approximate $\Omega G$ by a finite dimensional manifold but by an infinite dimensional one.

In slightly more detail, the idea of the proof is to find a Grassmannian model for $\Omega G$. This is done by considering the action of $G$ on $L^2(S^1;\mathfrak{g}_\mathbb{C})$ and then taking the restricted Grassmannian of this space. Within that, one can identify a sub-Grassmannian that is diffeomorphic to $\Omega G$. They then find a cell decomposition of this Grassmannian and analyse that. One of the important pieces is to consider the subgroup of polynomial loops in $\Omega G$. This corresponds to a certain sub-Grassmannian and it's easy to see that for this Grassmannian then all the cells are of even dimension, whence $\pi_1(\Omega_{\operatorname{pol}} G)$ is trivial. The final step is thus to show that the two Grassmannia (corresponding to $\Omega G$ and $\Omega_{\operatorname{pol}} G$) are homotopy equivalent. Then $\Omega G$ and $\Omega_{\operatorname{pol}} G$ are homotopy equivalent and so $\pi_1(\Omega G)$ is trivial.

Thus $\pi_2(G)$ is trivial since $\pi_2(G) = \pi_1(\Omega G)$ (incidentally, one doesn't need the long exact sequence for fibrations to see that $\pi_k(X) = \pi_{k-1}(\Omega X)$; that's either by definition or by using the adjunction $[\Sigma X, Y] \cong [X, \Omega Y]$).

As I said, Pressley and Segal say that this is in essence the same as Bott's proof; meaning that it proceeds by a cell decomposition based on "energy". However, it treats the infinite dimensional spaces as infinite dimensional spaces so I like it! Also, Grassmannia are more obviously structured so the cell decomposition may be simpler to see and understand in the Grassmannian model for the loop group.

Answer 6

This is proved in the book Representations of compact Lie groups by Bröcker and tom Dieck and reviewed in the Bulletin of the AMS. It is Proposition 7.5 in Chapter V. The proof is for compact, connected Lie groups, but any connected Lie group has the homotopy type of its maximal compact subgroup. (Everything here is for finite-dimensional Lie groups, of course.)

Edit: Maybe I should add, given that two of the other answers mention similar proofs, that the one in this book does not use Morse theory. It uses only basic covering space techniques once it is shown that $\pi_2(G)$ is isomorphic to $\pi_2(G_r)$, where $G_r$ are the regular elements, itself not a difficult lemma.

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Edit: The following is wrong! It is only true rationally, which is why I didn't remember having seen a proof of the general case :)

Also if you believe that simply-connected compact Lie groups have the homotopy type of a product of odd spheres, then this follows. It is fairly easy to see that this is the case rationally, but do not remember whether the general statement is hard to prove.

Answer 7

There is a sketch of the proof using the maximal torus as an exercise in Greub-Halperin-Vanstone's book.