A splitting theorem for good complexifications

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.