An Enlargeability Obstruction for Spacetimes with both Big Bang and Big Crunch

Given a spacelike hypersurface $M$ of a time-oriented Lorentzian manifold $(\overline{M}, \overline{g})$, the pair $(g, k)$ consisting of the induced Riemannian metric $g$ and the second fundamental form $k$ is known as initial data set. In this article, we study the space of all initial data sets $(g, k)$ on a fixed closed manifold $M$ that are subject to a strict version of the dominant energy condition. Whereas the pairs of the form $(g, \tau g)$ and $(g, -\tau g)$, for a sufficiently large $\tau > 0$, belong to the same path-component of this space when $M$ admits a positive scalar curvature metric, it was observed in a previous work \cite{arXiv:1906.00099} that this is not the case when the existence of a positive scalar curvature metric on $M$ is obstructed by $\alpha(M) \neq 0$. In the present article we extend this non-connectedness result to Gromov-Lawson's enlargeability obstruction, which covers many examples, also in dimension $3$. In the context of relativity theory, this result may be interpreted as excluding the existence of certain globally hyperbolic spacetimes with both a big bang and a big crunch singularity