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 M1 such that
M1 admits a fiber bundle structure with base (S1)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.Comment: 13 pgs no fig