2-groups and approximate fibrations

This paper strives for instant detection of approximate fibrations. A wide class of homotopy types sustains this effort, in the sense that those maps between manifolds for which all point preimages have the homotopy type of a fixed object in this class necessarily are approximate fibrations. Often, as exemplified here, the fundamental group or merely the first homology group carries the determining feature. The main result promises that the closed manifold N is a codimension-2 fibrator, meaning that it has the desired effect on proper maps from an (n+2)-manifold M onto a metric space B whose fibers all have the homotopy type of N, if π1(N) is an Abelian 2-group. Under more restrictive conditions, the same conclusion about N holds if either H1(N) is a cyclic 2-group or if H1(N) is an arbitrary finite cyclic group and no element of π1(N) has finite order.