Question

I wish to thank Professor Claudio Gorodski for his very helpful answers to my question on the webcite: http://mathoverflow.net/questions/67030/if-compact-connected-lie-groups-are-homeomorphic-as-topological-space-are-they-i

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?

Answer 1

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.

Rector, David L. Loop structures on the homotopy type of S3. Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971), pp. 99–105. Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971.