The theory by no means is only local.
The global degrees of freedom are determined by the curvature invariants integrated over all space. In other words if the local observables are determined everywhere, then topology can be discussed formally in terms of physical observables or better said: it's predetermined although inaccessible!
Take a look at the Chern-Gauss-Bonnet theorem for even dimensional Riemannian manifolds.
For example the Euler characteristic (a global topological feature) can be explicitly computed in terms of observables in case of Riemannian manifolds.
Another example is Sphere Theorem where enough knowledge about the curvature map(to be defined below) of the manifold fixes the topology.(although there's a mild topological assumption of simply connectedness here, but the topological information you are given at the end is way nontrivial in comparison to the input!)
Keep in mind that Riemann curvature tensor is the only physical observable that can be measured by means of geodesic deviation at every point of space.
But in case of Lorentzian manifolds, which is our case in relativity things become complicated.
There exist two topologies, one is that of the manifold topology and the other is the Alexandrov topology that is defined in terms of the Lorenzian metric. I like to call such topology the observable topology!
These two topologies do not necessarily coincide unless the spacetime is at least strongly causal on the causal ladder.
So in case the topologies coincide the global topology can be fixed in terms of what I call the curvature map: the map of curvature at all spacetime points.
But in case the two topologies do not coincide, the manifold topology is an independent degree of freedom that should be treated separately.
Nonetheless there are Pseudo-Riemannian variants of the Chern-Gauss-Bonnet theorem that determine the manifold topology partially, in terms of the observables: Chern-Gauss-Bonnet for Pseudo-Riemannian manifolds
So maybe the real question is: to what extent the manifold topology is restricted by the curvature map in the Lorentzian manifold that is not strongly causal?
PS: I always assumed that spacetime points are defined operationally by means of spacetime coincidences argument. Given such definition of spacetime, then a curvature map generally demands multiple compatible such observers.
PS: For those who might object that, the topological information can be localized by a singular gauge transformation(under the influence of the topological defects in gauge theory), one should note that such singularity in case of gravity is hidden behind a horizon(since this singularity is physical), and the information remains inaccessible to the external observer!