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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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Textbooks on equivariant homotopy theory, symmetry, etc, would be a start. –  Ryan Budney May 11 '11 at 10:28
    
A similar discussion mathoverflow.net/questions/42249/… –  Júlio César May 11 '11 at 21:02

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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.

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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.

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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. –  Júlio César May 11 '11 at 21:00

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