Constructing approximate fibrations

In this paper two results concerning the construction of approximate fibrations are established. The first shows that there are approximate fibrations $p:M \to S^2$ which are homotopic to bundle maps but which cannot be approximated by bundle maps. Here $M$ can be a compact $Q$-manifold or some topological $n$-manifold, $n \geqslant 5$. The second shows how to construct approximate fibrations $p:M \to B$ whose fibers do not have finite homotopy type, for any $B$ of Euler characteristic zero. Here $M$ can be a compact $Q$-manifold and $B$ only has to be an ANR, or $M$ can be an $n$-manifold, $n \geqslant 6$, and $B$ must then also be a topological manifold.