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