Abstract
All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from $\scri^-$ to ${\scri}^+$ is homotopic to a topologically trivial curve from $\scri^-$ to ${\scri}^+$. (If the Poincar\'e conjecture is false, the theorem does not prevent one from probing fake 3-spheres).
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