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, [$\pi\_2(G) = 0$](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups)


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](https://mathoverflow.net/questions/129/how-can-you-tell-if-a-space-is-homotopy-equivalent-to-a-manifold)



Homotopy properties of Lie groups
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](http://web.math.ku.dk/%7Ejg/papers/icm.pdf).


