Submanifold decompositions that induce approximate fibrations (Q1825479)

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.