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Does every compact connected manifold carry at most one continuous group structure? In other words, let G and G’ be compact connected Lie groups. If G and G’ are homeomorphic does that imply they are isomorphic as Lie groups?

Does uniqueness follow from classification of compact connected Lie groups? Even if so, is there a more direct way of showing uniqueness?

I know that uniqueness fails for non connected case ( for example, finite groups) and it fails for non compact case ( for example there are many distinct groups structures on Euclidean space for any $ n \geq 2$).

Ian Teixeira
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No. Consider, for instance, the group $U(2)$. It is homeomorphic to $SU(2)\times U(1)$. However, $U(2)$ and $SU(2)\times U(1)$ are not isomorphic Lie groups.

José Carlos Santos
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  • Does uniqueness hold for semi simple groups? – Ian Teixeira Mar 10 '20 at 22:04
  • No. The Lie groups $SO(4,\mathbb R)$ and $SO(3,\mathbb R)\times SU(2)$ are homeomorphic but not isomorphic. – José Carlos Santos Mar 10 '20 at 22:07
  • Complement: one way to see these homeomorphisms is to find semidirect decompositions $\mathrm{U}(2)=\mathrm{SU}(2)\rtimes\mathrm{U}(1)$ and $\mathrm{SO}(4)\simeq\mathrm{SU}(2)\rtimes\mathrm{SO}(3)$. – YCor Mar 10 '20 at 22:13
  • @IanTeixeira why "does that mean"? nobody said that. (And anyway have you computed the dimension?) – YCor Mar 10 '20 at 22:14
  • I was just trying to cook up a simply connected example so I took double covers of both sides from his example – Ian Teixeira Mar 10 '20 at 22:15
  • Sorry I meant spin 3 both times just a typo! I’ve edited my response to reflect that – Ian Teixeira Mar 10 '20 at 22:17
  • @IanTeixeira you mean 4 on the right too... – YCor Mar 10 '20 at 22:17
  • Oh yes wow sorry a lot of typos. My ability to edit the comment timed out so I’ll just write the correct version here: “ Does that mean $ spin(3)^2 \cong spin(4) $” – Ian Teixeira Mar 10 '20 at 22:18
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    Anyway homeomorphic simply connected compact Lie groups are isomorphic; I think this follows from classification. ($\mathrm{Spin}(3)^2$ is definitely isomorphic to $\mathrm{Spin}(4)$) – YCor Mar 10 '20 at 22:18
  • That’s great! In some ways the simply connected case is what I’m really interested in anyway. Could you explain further or give a reference for the fact that homeomorphic simply connected compact Lie groups are isomorphic? If there is anything I can do to help, like make a new question or edit this question, let me know! – Ian Teixeira Mar 10 '20 at 22:30