Mappings from $3$-manifolds onto $3$-manifolds

Abstract:

Let f be a compact, boundary preserving mapping from the 3-manifold ${M^3}$ onto the 3-manifold ${N^3}$. Let ${Z_p}$ denote the integers mod a prime p, or, if $p = 0$, the integers. (1) If each point inverse of f is connected and strongly 1-acyclic over ${Z_p}$, and if ${M^3}$ is orientable for $p > 2$, then all but a locally finite collection of point inverses of f are cellular. (2) If the image of the singular set of f is contained in a compact set each component of which is strongly acyclic over ${Z_p}$, and if ${M^3}$ is orientable for $p \ne 2$, then ${N^3}$ can be obtained from ${M^3}$ by cutting out of $\operatorname {Int} \;{M^3}$ a compact 3-manifold with 2-sphere boundary, and replacing it by a ${Z_p}$-homology 3-cell. (3) If the singular set of f is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of f are cellular.