Manifolds that induce approximate fibrations in the PL category

This paper provides quick recognition of approximate fibrations among certain PL maps by identifying fibrators. A closed connected orientable n-manifold N is called a codimension k PL fibrator if, for all PL maps p: M → B on a PL (n + k)-manifold M such that each p−1b collapses to a copy of N, p is an approximate fibration. As codimension 2 fibrators are fairly well understood, the paper emphasizes codimensions k > 2. Here are some of its key results, all stated for a codimension 2 PL fibrator N. If aspherical, N is a codimension 3 PL fibrator. If its universal cover is closed and (k − 1)-connected, k < 6, then N is a codimension k PL fibrator. If N is 3-dimensional and π1(N) ≠ Z2 ∗ Z2, then N is a codimension 3 fibrator. Again for 3-dimensional N, if π1(N) is normally cohopfian and either N is aspherical or H1(N) is infinite, then N is a codimension 4 PL fibrator. If N is aspherical, π1(N) is normally cohopfian, and π1(N) has no proper normal subgroup isomorphic to π1(N)A where A itself is an Abelian normal subgroup of π1(N), then N is a codimension k PL fibrator for all k. In particular, the final statement holds for all hyperbolic 3-manifolds as well as for those with two other geometric structures.