Linking, causality and smooth structures on spacetimes (based on joint work with Stefan Nemirovski).

Abstract:  Globally hyperbolic spacetimes form probably the most important class of spacetimes. Low conjecture and the Legendrian Low conjecture formulated by Nat\’ario and Tod say that for many globally hyperbolic spacetimes X two events x,y in X are causally related if and only if the link of spheres S_x, S_y whose points are light rays passing through x and y is non-trivial in the contact manifold N of all light rays in X. This means that the causal relation between events can be reconstructed from the intersection of the light cones with a Cauchy surface of the spacetime.

We prove the Low and the Legendrian Low conjectures and show that similar statements are in fact true in almost all $4$-dimensional globally hyperbolic spacetimes. This also answers the question on Arnold’s problem list communicated by Penrose.

We also show that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric, thus global hyperbolicity imposes censorship on the possible smooth structures on a spacetime. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R^4.