Homotopy equivalences on $3$-manifolds

Abstract:

Suppose ${M^3}$ is a $3$-manifold and $f:{M^3} \to X$ 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 $\alpha$ of $X$ there exists an open cover $\beta$ of $X$ such that if $f$ is a $\beta$-equivalence then each ${f^{ - 1}}\left ( x \right )$ is $\alpha$-cellular in $M_ + ^3 \times {{\mathbf {R}}^1}$. In fact, the (open) cellularity occurs in a continuous fashion and so the map $f$ can be approximated by a Euclidean bundle map.