Cosmic censorship of smooth structures and integer valued metrics on spacetimes

The talk is based on joint results with Stefan Nemirovski.
We show that absence of time travel and naked singularities imposes strong restrictions on the smooth structure of a 4-dimensional spacetime. In particular every such contractible spacetime is diffeomorphic to the standard $R^4$. Also every such spacetime homeomorphic to $M\times R$ with $M$ closed oriented 3-manifold is diffeomorphic to $M\times R$.
We introduce integer valued semi-metrics on spacetimes, motivated by the work of Collin and Sandon on Legendrian manifolds. These semimetrics are metrics exactly when spacetime does not have time travel and they are zero for totally vicious spacetimes.