The PL fibrators among aspherical geometric 3-manifolds (Q1345496)

Let \(N\) be a closed, connected, orientable PL \(n\)-manifold. In the author's terminology, \(N\) is a codimension-\(k\) PL fibrator if every proper PL map \(p:M\to B\) with \(M\) a connected orientable PL \((n+k)\)-manifold, \(B\) a polyhedron, and all fibers \(p^{-1}(b)\) \(N\)-like, is an approximate fibration; here, a polyhedron is said to be \(N\)-like if it collapses to an \(n\)-dimensional subpolyhedron homotopy equivalent to \(N\). The author has demonstrated, in a series of papers, that PL fibrators are not exceptions but rather commonplace. Best understood are codimension-1 and codimension-2 PL fibrators. The three main theorems of the present paper give conditions under which a codimension-2 PL fibrator is also a PL fibrator in higher codimensions. The most important, and the easiest to state, are the following two: 1. If the universal covering of \(N\) is compact and \((k-1)\)-connected \((k>2)\), then \(N\) is a codimension-\(k\) PL fibrator if and only if it is a codimension-2 PL fibrator. 2. Suppose \(n=3\) and \(N\) is aspherical and virtually geometric (i.e., some finite covering of \(N\) is a geometric 3-manifold). Then \(N\) is a PL fibrator in all codimensions if and only if it is a codimension-2 PL fibrator. Other results of this paper give sufficient conditions, mainly in terms of \(\pi_1(N)\), for \(N\) to be a PL fibrator.