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Submanifold decompositions that induce approximate fibrations. (English) Zbl 0684.57009
A closed n-manifold N is called a codimension-k fibrator if for every usc decomposition G of an $$(n+k)$$-manifold M such that each $$g\in G$$ is shape equivalent to N and the decomposition space M/G is finite dimensional, then the projection $$\pi$$ : $$M\to M/G$$ is an approximate fibration. The main problem studied in this paper is the identification of codimension-k fibrators. As a sample, we state the following results proved in this paper:
(1) A closed orientable n-manifold N is a codimension-1 fibrator provided one of the following is satisfied: (a) $$Z\pi_ 1(N)$$ is Noetherian; or (b) $$\pi_ 1(N)$$ is Hopfian and N is aspherical.
(2) A closed n-manifold N is a codimension-2 fibrator provided one of the following is satisfied: (a) N is the real projective n-space $$(n>1)$$; (b) N is a surface, $$n=2$$, and the Euler characteristic of N is nonzero; or (c) each element of $$\pi_ 1(N)$$ has order two. Several examples, counterexamples, and relevant questions are also given.
Reviewer: S.Singh

##### MSC:
 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 55R65 Generalizations of fiber spaces and bundles in algebraic topology 57N25 Shapes (aspects of topological manifolds) 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 54B15 Quotient spaces, decompositions in general topology 57N10 Topology of general $$3$$-manifolds (MSC2010)
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