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.

nothomeomorphic to S^2. I know because I made the same mistake a few months ago! The intuitive reason that SO(3) and S^2 can't be homeomorphic is that SO(3) is a 3-dimensional manifold, while S^2 is 2-dimensional. To see why SO(3) is 3-dimensional, imagine a sphere with a dot on it; hold it so the dot is on top. There are three independent ways to rotate the sphere. First, you can move the dot towards or away from your body. Second, you can move the dot to the left or the right. Third, you can spin the sphere around the dot, so the dot doesn't move. $\endgroup$10more comments