Proceedings of the American Mathematical Society
Suppose M3 is a 3-manifold and f: M3 ⟶ X is a homotopy equivalence onto an ANR X. In this paper the cellularity properties of point preimages under f are studied. It is shown that for every open cover α of X there exists an open cover β of X such that if f is a β-equivalence then each f-1(x) is α-cellular in M3 X R1. In fact, the (open) cellularity occurs in a continuous fashion and so the map f can be approximated by a Euclidean bundle map.
Venema, Gerard A., "Homotopy equivalences on 3-Manifolds" (1984). University Faculty Publications and Creative Works. 553.