lie groups and topology

Question

Is there a relationship between Lie groups and topology and is there a succinct explanation that can be provided? Is there a good online reference that discusses this.

Answer 1

The short answer is yes. Lie groups form an important class of examples of topological spaces with interesting topological properties. One famous example is Bott periodicity, which is a calculation of the stable homotopy groups of certain classes of Lie groups.

Answer 2

One could define the vast topic of "functional analysis" to be about topological spaces that also have an algebraic structure, such that there is a relation like "certain algebraic operations are continuous". Lie groups are an example of this, they combine a topological structure (being a topological manifold) with an algebraic structure (being a group) such that the group operations are continuous.

This simple relation already has profound implications, one of which has been the topic of Hilbert's fifth problem.