# Are there nonsmooth Lie groups?

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.

• Continuous meaning? Not-discrete? – Cameron Williams May 28 '15 at 0:37
• I think he means a topological manifold. – shalop May 28 '15 at 0:41
• 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. – Ben Grossmann May 28 '15 at 0:44
• Yes, topological. Fixed. – Geoffrey Irving May 28 '15 at 0:44
• 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. – Geoffrey Irving May 28 '15 at 0:48