# zbMATH — the first resource for mathematics

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

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 55R65 Generalizations of fiber spaces and bundles in algebraic topology