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Let $G_{1}$ and $G_{2}$ be compact connected Lie groups.

If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are they isomorphic as Lie groups?

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2 Answers 2

No. The simplest example I can think of is that $SO(4)$ is homeomorphic to $SO(3)\times Sp(1)$ as topological spaces, but they are not isomorphic Lie groups. In fact, there is a double covering $Sp(1)\times Sp(1)\to SO(4)$, $(q_1,q_2)\cdot x = q_1xq_2^{-1}$, where $Sp(1)$ is viewed as the unit quaternions and $x\in\mathbf H\cong\mathbf R^4$, which gives a Lie group isomorphism $Sp(1)\times Sp(1)/\{\pm1\}\cong SO(4)$.

On the other hand, every continuous homomorphism between Lie groups is automatically smooth, so if the homeomorphism is a homomorphism, then it must be a Lie group isomorphism.

Edit: I was puzzled by the fact that all known examples are given by pairs of locally isomorphic groups. Then I found the following papers:

Toda, H., A note on compact semi-simple Lie groups, Japan J. Math. 2 (1976), 355-358.

in which he proves that "two simply connected, compact (and hence semi-simple) Lie groups are isomorphic to each other if and only if they have isomorphic homotopy groups for each dimension". Later this was generalized by S. Boekholt (Journal of Lie Theory Volume 8 (1998) 183-185) as follows:

"Let $G_1$ and $G_2$ be two compact, connected Lie groups with isomorphic homotopy groups in each dimension. Then $G_1$ and $G_2$ are locally isomorphic."

So, going back to the question, we cannot avoid local isomorphism.

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No.

For instance, topologically $U(2) = SU(2) \times U(1)$, since both are homeomorphic to $S^3 \times S^1$, but the group structures are different. Another example is given by $SO(3) \times SU(2)$ which is diffeomorphic to $SO(4)$.

On the other hand, any commutative connected compact real Lie group of dimension $n$ is isomorphic (as a real Lie group) to the real torus $\mathbb{T}^n:=(\mathbb{S}^1)^n$.

Analogously, any connected compact complex Lie group of dimension $n$ is isomorphic (as a complex Lie group) to a complex torus, i.e. a quotient of the form $\mathbb{C}^n/\Gamma$, where $\Gamma \subset \mathbb{C}^n$ is a lattice. Notice that in the compact complex case commutativity comes for free. Two complex tori $\mathbb{C}^n /\Gamma_1$ and $\mathbb{C}^n / \Gamma_2$ are isomorphic as complex Lie groups if and only if there exists $g \in \textrm{GL}_n (\mathbb{C})$ such that $\Gamma_2=g (\Gamma_1)$, but of course they are always both isomorphic to $\mathbb{T}^{2n}$ as real Lie groups.

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