Euler characteristic of Cauchy surface in Lorentz manifold Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics. 
If the above question is too general, I am specifically interested in the two subcases:


*

*What if we require the Cauchy hypersurface to be compact?

*And/or what if we require the ambient manifold to be flat?


Motivation
(Copied from comments below)
I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d (2 space and 1 time) I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.
I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including ones with compact Cauchy hypersurfaces (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.
 A: I read your question as the one about compact Cauchy surfaces in locally flat space times. Then the answer is negative: Take quotient of upper hyperboloid $H$ in $R^{2,1}$ (i.e. the hyperbolic plane) by a torsion free discrete cocompact subgroup $\Gamma$ in $SO(2,1)$. The Euler characteristic will be negative. Now, take the future light cone $C$ in $R^{2,1}$ and take the quotient $C/\Gamma$. This is your locally flat space-time, containing $H/\Gamma$ as a Cauchy hypersurface. This manifold is, of course, incomplete, but, if I remember correctly, Geoff Mess proved that you cannot have a compact Cauchy hypersurface in a complete locally flat Lorentzian manifold. 
Edit: On the other hand, there are topological obstructions for existence of locally flat Lorenztian metrics. For instance, suppose that $M$ is a compact n-dimensional simply-connected manifold which does not immerse in $R^{n+1}$. For instance, $CP^2$ is an example of such manifold. Then $M\times R$ does not admit a locally flat Lorentzian metric, even incomplete one. (This follows because the developing map to $R^{n,1}$ of such a structure would yield an immersion to $R^{n,1}$.  
Talking about tiles: My guess is that you are actually trying to construct complete locally flat Lorentzian metrics. It is known (proved by Geoff Mess) that such metrics do not exist on manifolds of the form $M\times R$, where $M$ is compact and hyperbolic. I will check, by now there is probably a reasonably good description of $M$'s for which such metric exists. On the other hand, if you take $M$ which is a noncompact surface, then such a metric does exist; these are so called Margulis space-times. One even has some nice description of their fundamental domains (tiles) due to Todd Drumm.  The trick is that the tiles are not convex. 
(Google "crooked planes" to learn more about them.) I am not sure if $M$ will be a Cauchy hypersurface in this case, I would have to check. 
See e.g. here for a somewhat dated survey. 
