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I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found references on:

  1. Lie Groups, An Approach through Invariants and Representations, by Claudio Procesi: page 80.

  2. Probability on Compact Lie Groups by David Applebaum: page 5.

But I have not found the actual statement nor proof of the result anywhere. Does anybody know where I may find it?

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  • $\begingroup$ I'm sure you've figured it out by now, but if you know the fact for topological groups, checking smoothness of multiplication and inversion should be the least "nightmarish" part because the covering space gets its smooth structure from the space it's covering. The covering map is a local diffeomorphis, by definition, and smoothness is a local property. $\endgroup$
    – Tim kinsella
    Jun 8, 2015 at 22:09

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It's a standard theorem in covering space theory, which you should be able to find in most introductory algebraic topology textbooks, that a space has a universal cover if it is path-connected, locally path-connected, and semi-locally simply connected. This is, for example, Theorem 82.1 in Munkres' Topology (2nd edition) and Proposition 1.36 in Hatcher's Algebraic Topology.

In particular, every connected manifold satisfies this condition, so every connected manifold has a universal cover (which is a simply connected manifold).

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  • $\begingroup$ Thank you! I knew about the theorem, I was looking for a more direct/basic way of proving it without using heavy artillery, but I'm failing miserably (I've managed to do so for topological groups, but checking that the multiplication and inverse are differentiable is a nightmare). $\endgroup$
    – Pablo S. Ocal
    Jun 4, 2015 at 7:29

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