Homotopy properties of Lie groups

Question

Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds

For example

1. $G$ is homotopy equivalent to a smooth compact orientable manifold. In particular, Poincaré duality holds for $G$.
2. $\pi_1(G)$ is abelian, $\pi_2(G) = 0$

The impossibly perfect answer to my question is a list of properties that make up a complete homotopy characterization of Lie groups (that is, in every homotopy type (of smooth manifolds) with such properties, there exists a smooth manifold admitting the structure of Lie groups).

P.S. In this question, I am not interested in the homotopy properties of manifolds that distinguish them from other CW-complexes, for this see resp. question on MO

Answer 1

The problem you mention has a long history. The best homotopy characterization is probably using the notion of finite loop spaces:

A finite loop space is a space $BG$ such that $\Omega BG$ is homotopy equivalent to a finite CW-complex. There are many of those, but one can give a precise homotopy characterization of which ones come from compact Lie groups: They are the ones admitting a "maximal torus", defined to be a map $({\mathbb C}P^\infty)^r \to BG$, such that the homotopy fiber is homotopy equivalent to a finite complex.

This was the so-called maximal torus conjecture, solved as a consequence of the classification of $p$-compact groups, which states that there is a 1-1-correspondence between connected $p$-compact groups and ${\mathbb Z}_p$-root data, parallel to the classification of connected compact Lie groups, but with $\mathbb Z$ replaced by the $p$-adic integers ${\mathbb Z}_p$.

As a consequence one can also give a classification of all finite loop spaces: If you pick a connected $p$-compact group for every prime $p$ agreeing over $ \mathbb Q$, there is an explicit double coset space of finite loop space structures with this $p$-local data.

A reference is my ICM survey The Classification of p–Compact Groups and Homotopical Group Theory.

Answer 2

A very useful fact is that every connected Lie group is rationally homotopy equivalent to the product of several odd dimensional spheres where the number of spherical factors is equal to the rank of the group. For example, $SU(n)$ is rationally equivalent to $S^3\times S^5 \ldots \times S^{2n-1}$ and $Sp(n)$ is rationally equivalent to $S^3\times S^7\times \ldots \times S^{4n-1}$. For simple Lie groups the first factor is always $S^3$ and there is only one $S^3$ in each simple group. Also, every simple $G$ has $\pi_3\cong\mathbb Z$. For reference see for example the book "Topology of transitive transformation groups" by Onishchik.