Rationality of geometric signatures of complete 4-manifolds

According to [CG4]|and [CFG], the complete manifolds with bounded sectional curvature and finite volume admit positive rank F-structures near infinity. In this paper, we show that, in dimension four, if the manifolds also have bounded covering geometry near infinity, then there exist F-structures with special topological properties. F-structures with these properties cannot be constructed solely by means of the general methods in [CG4]|and [CFG]. Using these special properties we prove a conjecture of Cheeger-Gromov on the rationality of the geometric signatures in the four dimensional case.