3-manifolds with geometric structure and approximate fibrations.

*(English)*Zbl 0731.57006
Indiana Univ. Math. J. (to appear).

Let p: \(M\to B\) be a proper map defined on an orientable 5-manifold M such that each \(p^{-1}b\) is homeomorphic to a fixed closed, orientable 3-manifold N. This paper investigates the geometric 3-manifolds N for which p is invariably an approximate fibration; more explicitly, its aim is to determine whether the presence of a particular geometric structure on N causes p to be one. Many of the manifolds N with the structure of \(S^ 3\) are known to have this approximate fibration-inducing feature; the majority of those with \(E^ 3\) and with \(H^ 2\times R\) structures do not, nor do the two with \(S^ 2\times R\) structure. New results include: (1) all hyperbolic, Sol, and \(SL_ 2(R)\) 3-manifolds induce approximate fibrations in this setting; and (2) some, but not all, Nil manifolds that fiber over \(S^ 1\) have the same feature, as do the other Nil manifolds that fail to fiber over \(S^ 1\).

Reviewer: R.J.Daverman