Surprising results about the geometry and topology of Lie groups

Question

I am new to the study of Lie groups, and I would like to know beforehand what are the most surprising and important results in this field of study. For example, as a person especially interested in algebraic topology, I find this post extremely interesting. What other results describe the particular geometry and topology of these objects?

Answer 1

One of my favourite results is a description of the cohomology of a compact Lie group in terms of invariants in the exterior algebra of its lie algebra.

To be more precise:

Let $G$ be a compact Lie Group and let $g$ be its Lie algebra (that can be viewed as the algebra of left translation invariants vector fields on $G$).

Conjugation induces an action of $G$ on $g$. If you identify $g$ with the tangent plane at $e$ of $G$, this action extends to a degree preserving action on $\Lambda g$, i. e. the space of left invariant differential forms.

A very famous theorem due to Cartan and proved in a more Lie-theoretic flavour by Reeder (see M.Reeder, "On the cohomology of compact Lie groups", L'Enseignement Math., t. 41, (1995), pages 181-200 ) , asserts that

$$H^*(G) \simeq (\Lambda g )^G$$

as graded vector spaces, where $(\Lambda g )^G$ is the ring of invariant differential forms for the above described action.

Moreover, a beautiful and instructive overview about the XX century topological discoveries concerning Lie groups can be found in the very illuminating

A. Borel "Topology of Lie groups and characteristic classes." Bull. Amer. Math. Soc.,Vol. 61, number 5 (1955), 397-432