A Quick Primer Physicists assume that the universe can be described as a manifold. Mathematician characterize manifolds in terms of their geometry and topology.
Geometry is a local quantity that measures the intrinsic curvature of a surface.
General relativity relates the mass distribution of the universe to its geometry and of course, the geometry of the universe determines the dynamics of the mass. General relativity does not at all constrain the topology of the universe. The relationship between topology and geometry is most familiar in flat space. There are several different ways of thinking about the topology of a manifold. It is simplest to begin by considering topology in flat two dimensional space.