Poincare's Theorem says that any simply-connected 3-dimensional manifold is a 3-sphere. But I'm not sure that it has been proven that the universe is simply-connected (any loop can be smoothly shrunk to a point). Perhaps the universe has the structure of a more complex 3-dimensional manifold.
That’s right. The proof was just that. It was proved to the satisfaction of topologists. That wouldn’t be me.