I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian.

I wonder if 'homeomorphic to a torus' is enough to imply abelian.

Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

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    $\begingroup$ Some thoughts: Every connected aspherical compact connected lie group is diffeomorphic to a torus, so if you find a connected compact aspherical non-abelian lie group, you are done. Every solvable lie group is aspherical, but this class doesn't give us anything, since every connected solvable compact lie group is already a torus (as a lie group). $\endgroup$ – archipelago Jul 4 '15 at 7:38
  • $\begingroup$ It is also enough to find a compact connected non-abelian lie group which admits a complete metric with nonpositive curvature, since all of those are aspherical. However I could not find an example immediately and I'm not even sure such a thing exists. $\endgroup$ – archipelago Jul 4 '15 at 7:40

In the compact case, the cohomology of a lie group coincides with the cohomology of it's lie algebra in particular $H^1(\frak g ) = \frak g / [ \frak g , \frak g ] $, since this has rank n for an n torus we have $\dim [ \frak g , \frak g ] = 0 $, so $G$ must be abelian.


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