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.
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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.
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.