3-MANIFOLDS WITH GEOMETRIC STRUCTURE AND APPROXIMATE FIBRATIONS

Let p: M --> B be a proper map defined on an orientable 5-manifold M such that each p-1b is homeomorphic to a fixed closed, orientable 3-manifold N. The purpose of this paper is to determine those geometric 3-manifolds N for which p is invariably an approximate fibration; more explicitly, it is to see whether the presence of a particular geometric structure on N causes p to be one. Many of the manifolds N with the structure of S3 are known to have this approximate fibration-inducing feature; the majority of those with E3 and with H-2 X R structures do not, nor do the two with S2 x R structure. New results include: (1) all hyperbolic, Sol, and SL2(R) 3-manifolds induce approximate fibrations in this setting; and (2) some, but not all, Nil manifolds that fiber over S1 have the same feature, as do the other Nil manifolds that fail to fiber over S1.