Submanifold decompositions that induce approximate fibrations.

*(English)*Zbl 0684.57009A 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.

(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) |

##### Keywords:

fundamental group; group ring; aspherical manifold; upper semicontinuous decomposition; closed n-manifold; codimension-k fibrator; shape equivalent; approximate fibration; codimension-1 fibrator; Noetherian; Hopfian; codimension-2 fibrator
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\textit{R. J. Daverman}, Topology Appl. 33, No. 2, 173--184 (1989; Zbl 0684.57009)

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