If local geometry is truly flat everywhere, Einstein field equations (Riemann tensor = 0) force global topology to be trivial in matter-free regions by topological rigidity theorems.

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Einstein Field Equations(General relativity): A single linked system of ten generally covariant partial differential equations from which the gross mechanical properties of bodies, comprising all gravitational-inertial phenomena, can be derived, and according to which spacetime and matter stand in dynamical interaction
Flat (geometry)(in physics and mathematics): Space that follows the rules of regular Euclidean geometry—like a piece of paper—rather than being curved or bent.
Global topology(as used in mathematics): The overall shape and structure of a space when you look at it as a whole, including how its different parts connect in the big picture.
Local geometry(in physics and mathematics): The shape and properties of space in a small, specific area around a point, like how a tiny patch of Earth's surface looks flat even though Earth is round.

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