This eloquently written paper presents further investigation of the problem when a given PL map is an approximate fibration. Let \(M\) be a connected, orientable, PL \((n + k)\)-manifold, \(B\) a polyhedron. A PL map \(p : M \to B\) is said to be \(N\)-like, where \(N\) is a fixed orientable \(n\)-manifold, if each \(p^{-1} b\) collapses to an \(n\)-complex homotopy equivalent to \(N\). \(N\) is called a codimension \(k\) PL fibrator, if for every orientable \((n + k)\)-manifold \(M\) and \(N\)-like \(p : M \to B\), \(p\) is an approximate fibration. \(N\) is a PL fibrator if it has this property for all \(k > 0\). E.g. previous results say that 2-manifolds, all but the 2-sphere and the torus are PL fibrators, and that also \((k - 1)\)-connected manifolds \((k > 1)\) are codimension \(k\) fibrators. The main result is that the complex projective \(n\)-space \(\mathbb{C} P^n\) is a codimension \(2n + 2\) PL fibrator. Concerning the proof one should point out that it employs quite a lot of algebraically topological arguments and in part depends on some previous results of the same author.