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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 coccompact 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.