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
- $G$ is homotopy equivalent to a smooth compact orientable manifold. In particular, Poincaré duality holds for $G$.
- $\pi_1(G)$ 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
Best Answer
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.