I should clarify. What I mean to ask is: if we are to think of Lie groups both as groups and manifolds, what conditions do we place on the manifold structure of a given group? To clarify the second paragraph: it is my understanding that if we make $SO(3)$ into a Lie group by thinking of it as a manifold as well as a group we can make the manifold structure homeomorphic to $S^2$. Could we make $SO(3)$ into a Lie group by thinking of it as some other manifold, what conditions do we impose on the topology when we do so? If this still isn't precise let me know and I will try to clarify.