The definition of "Lie group" typically restricts to a smooth manifold. If we instead define a "Lie group" to be a topological manifold such that multiplication and inversion are continuous, is the manifold necessarily smooth? Is the smooth structure unique if we want a smooth Lie group?

I believe the answer is yes, since a connected Lie group structure is determined by the Lie algebra, but my search attempts failed.

  • $\begingroup$ Continuous meaning? Not-discrete? $\endgroup$ – Cameron Williams May 28 '15 at 0:37
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    $\begingroup$ I think he means a topological manifold. $\endgroup$ – shalop May 28 '15 at 0:41
  • $\begingroup$ Careful: if you have a topological manifold, then we might not have a "tangent space", and so we wouldn't have a tangent space at the identity, and so the "Lie group" wouldn't have an associated Lie algebra. $\endgroup$ – Ben Grossmann May 28 '15 at 0:44
  • $\begingroup$ Yes, topological. Fixed. $\endgroup$ – Geoffrey Irving May 28 '15 at 0:44
  • $\begingroup$ The Lie algebra comment is intuition only: in the smooth case, the entire structure is determined by any neighborhood. That much is certainly also true in the topological case, but you're right that it's not obvious something more baroque than a Lie algebra might not be needed. $\endgroup$ – Geoffrey Irving May 28 '15 at 0:48

A Lie group is by definition a group internal to the category of smooth manifolds. So it doesn't make sense to ask for non-smooth Lie groups, just like how it doesn't make sense to ask for a ring which does not have an underlying abelian group.

The right question to ask (which I think is the gist of your question) is how to tell when a topological group has a unique smooth structure that makes it into a Lie group. One answer to this is that every locally compact and locally contractible topological group has a unique Lie group structure (Hofmann-Neeb arXiv:math/0609684). In particular, this means that if a topological group is known to be a topological manifold, then it has a unique Lie group structure.

This MO question discusses this and other facts.

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    $\begingroup$ Thank you for the answer. I am aware of the definition of a Lie group, as indicated in the question text. I apologize if the subject was less clear. $\endgroup$ – Geoffrey Irving May 28 '15 at 1:14

See also: Hilbert's Fifth Problem...


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