# On homeomorphic compact connected topological groups

I wish to thank Professor Claudio Gorodski for his very helpful answers to my question on the webcite: If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?

He said: Let $G_{1}$ and $G_{2}$ be two compact, connected Lie groups with isomorphic homotopy groups in each dimension. Then their Lie algebras are isomorphic.

Now my question is: If $G_{1}$ and $G_{2}$ are two compact, connected topological groups which are homeomorphic as topological space, are there any isomorphism theorems?

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Haven't we had this one before? I think the $p$-adic integers (for different $p$) was the example given -- all of these spaces are homeomorphic IIRC but for different $p$ they're not at all isomorphic as topological groups. I'm going from memory here so apologies if I'm off base. – Kevin Buzzard Jun 7 '11 at 6:59
.. or even as abstract groups. – algori Jun 7 '11 at 7:01
What is IIRC? Is your example a connected topological space? – sife Jun 7 '11 at 7:42
sife -- iirc means "if I remember correctly". iirc, that is.. – algori Jun 7 '11 at 9:23
@Kevin - This question? mathoverflow.net/questions/44060/… – Steven Gubkin Jun 7 '11 at 14:05

If you replace homeomorphic with homotopy equivalent the answer is no. There are infinitely-many non-isomorphic topological groups which are homotopy equivalent (just as spaces) to $S^3$. Actually, these topological groups cannot be connected by a zig-zag of group homomorphisms which are homotopy equivalences. But of course all of them have the same homotopy groups.
Fernando Muro--Are the examples mentioned above compact, connected topological groups? If they are compact, connected topological space, then they must be homeomorphic to $S^{3}$ by Poincaré conjecture . Therefore we need not replace homeomorphic with homotopy equivalent. – sife Jun 7 '11 at 16:00