What is a local group isomorphism?

Question

What does it mean for 2 groups to be locally isomorphic?

E.g. $SO(4)$ is locally isomorphic to $ SO(3)\times SO(3)$ -why not globally?

Answer 1

The answer is: for groups it means nothing.

However, two topological groups $G,H$ are said to be locally isomorphic if there are open neighborhoods $V_G$, $V_H$ of $1_G$ and $1_H$ and a homeomorphism $f:V_G\to V_H$ such that for all $x,y\in V_G$, we have $xy\in V_G$ $\Leftrightarrow$ $f(xy)\in V_H$, and if so, then $f(xy)=f(x)f(y)$.

Classical theorems (see for instance Bourbaki, Lie groups and algebras, Chap 2) are that for Lie group over $\mathbf{R}$ or $\mathbf{Q}_p$ (or a product thereof, such as the solenoid $(\mathbf{R}\times\mathbf{Q}_p)/\mathbf{Z}[1/p]$), to be locally isomorphic is equivalent to having isomorphic Lie algebras.

Also, two complex Lie groups are locally isomorphic if there is $f$ as above that is, in addition, biholomorphic. This is also characterized as having isomorphic complex Lie algebras. It is stronger than being locally isomorphic as topological groups.

Answer 2

For Lie groups specifically what should happen is that two Lie groups are locally isomorphic iff they have isomorphic Lie algebras. $SO(4)$ and $SO(3) \times SO(3)$ are locally isomorphic because they have isomorphic Lie algebras

$$\mathfrak{so}(4) \cong \mathfrak{so}(3) \times \mathfrak{so}(3)$$

or, said another way, because the have isomorphic universal covers

$$\text{Spin}(4) \cong \text{Spin}(3) \times \text{Spin}(3)$$

(see spin group). This is an example of an exceptional isomorphism.