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A parallel lightlike vector field on a Lorentzian manifold $X$ naturallydefines a foliation $\mathcal{F}$ of codimension one. If either all leaves of$\mathcal{F}$ are compact or $X$ itself is compact admitting a compact leaf andthe (transverse) Ricci curvature is non-negative then a Bochner type argumentimplies that the first Betti number of $X$ is bounded by $1 \leq b_{1} \leq\dim X$ if $X$ is compact and $0 \leq b_{1} \leq \dim X -1$ otherwise. We showthat these bounds are optimal and depending on the holonomy of $X$ we obtainfurther results. Finally, we classify the holonomy representations for those$X$ admitting a compact leaf with finite fundamental group.

Riemannian Foliations and the Topology of Lorentzian Manifolds