I'm trying to get a better handle on the relation between Lie groups and the Manifolds they correspond to. Firstly, is the relationship injective? that is, does each Lie group correspond to a unique manifold? Or are all the manifolds corresponding to a particular group homeomorphic?


Also, what formal form does the relationship take? I can intuitively understand the relationship between, say, $SO(3)$ and $S^2$ by thinking about rotating the sphere into itself, but what how does this generalize to a more general group or manifold.



Lie Groups and Manifolds
To add a bit,


There are also many examples of compact manifolds with multiple group structures.


As a quick example, first recall that $SU(2)$ is the collection of all $A \in M\_2(\mathbb{C})$ with $A\overline{A}^t = Id$ and $det(A) = 1$. It is a Lie group (which is actually diffeomorphic to $S^3$.)


The manifold $S^1\times SU(2)$ has (at least) 2 group structures. The first is simply the product. The second is isomorphic to the Lie group $U(2)$, those matrices $A\in M\_2(\mathbb{C})$ such that $ A\overline{A}^t = Id$ (no extra condition no the determinant).


For another example, recall that $SO(n)$ is the Lie group consisting of all $A\in M\_n(\mathbb{R})$ such that $AA^t = Id $. Then $SO(3)\times SU(2)$ is diffeomorphic to $SO(4)$ but the group structures are different.


