This talk is based on joint work with Stefan Nemirovski. Globally hyperbolic spacetimes form probably the most important class of spacetimes. Low conjecture and the Legendrian Low conjecture formulated by Natário 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 R⁴.