1
$\begingroup$

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.

  • 0
    A similar discussion http://mathoverflow.net/questions/42249/how-do-lie-groups-classify-geometry2011-05-11

2 Answers 2

4

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.

0

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.

  • 0
    Don´t you mean "Lie groups are an example of this, they combine a topological structure (being a smooth manifold) with an algebraic structure (being a group) such that the group operations are smooth."? Otherwise we would be simply talking about topological groups.2011-05-11