Manifolds that induce approximate fibrations in the PL category (Q1903003)

A map \(f : M \to B\) is called an approximate fibration if it has an approximate homotopy lifting property. The famous example of an approximate fibration is a cell-like map between manifolds. The author considers PL maps of manifolds \(f : M \to B\) with a fixed fiber \(N\) and studies the situation when such maps are approximate fibrations. When \(N\) is a manifold the answer depends on the topology of \(N\) and on the codimension: \(\dim M - \dim N\). Thus, it is proven that if \(N\) is an aspherical manifold and \(\text{codim} N = 3\), then \(f\) is always an approximate fibration. The asphericity is essential here, since there is a counterexample for \(N = S^2\). The paper contains many results for the case of 3-dimensional \(N\). The answer there depends on the fundamental group of \(N\). In the case of hyperbolic \(N\), every \(N\)-like map \(f : M \to X\) of any codimension is an approximate fibration.