Approximate fibrations and a movability condition for maps

In a previous paper the authors defined the approximate homotopy lifting property and studied its implications. This property is a generalization of the homotopy lifting property of classical fiber space theory. Here a necessary and sufficient condition on point-inverses for a map to have the approximate homotopy lifting property for %-cells is given; and the approximate homotopy lifting property for w-cells is shown to imply the approximate homotopy lifting property for all spaces. A corollary is that, in a fairly general context, any two point-inverses of a Serre (weak) fib rat ion have the same shape. By combining these results with results of L. Husch, some conditions are obtained under which a map between manifolds can be approximated by locally trivial fibrations.