  The purpose of this paper is to produce restrictions on fundamental groups of
manifolds admitting good complexifications by proving the following
Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a
good complexification has a finite-sheeted regular covering $M_1$ such that
$M_1$ admits a fiber bundle structure with base $(S^1)^k$ and fiber $N$ that
admits a good complexification and also has zero virtual first Betti number. We
give several applications to manifolds of dimension at most 5.
