  A parallel lightlike vector field on a Lorentzian manifold $X$ naturally
defines 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 and
the (transverse) Ricci curvature is non-negative then a Bochner type argument
implies 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 show
that these bounds are optimal and depending on the holonomy of $X$ we obtain
further results. Finally, we classify the holonomy representations for those
$X$ admitting a compact leaf with finite fundamental group.
