Necessary and sufficient conditions for codimension-$k$ maps to be approximate fibrations

Abstract

Let $N$ be a closed $n$-manifold with residually finite, torsion free $\pi_1(N)$ and finite $H_1(N)$. Suppose that $\pi_k(N)=0$ for $1 < k < n-1$. We show that $N$ is a codimension-$n$ PL fibrator if and only if $N$ does not cover itself regularly and cyclically up to homotopy type, provided $\pi_1(N)$ satisfies a certain condition.