Let $X$ and $Y$ be triangulate manifolds. It is clear that all triangulable manifolds can be given a regular CW decomposition.
For which continuous functions $f:X \to Y$ can we find regular decompositions so that $f$ is cellular?
I understand that by the cellular approximation given the decompositions we may find a $g$ homotopic to $f$ which is cellular, but I can't see how this would help.
$\textit{Sketch proof}$: Let $f:X \to Y$ be a branched cover function between $n$-manifolds. The branch set $B_{f} \subseteq X$ is the set of points where $f$ fails to be a local homeomorphism.
P. T. Church proved that $\dim(B_{f}) = \dim (f(B_{f})) \leq n-2 $.
$\textbf{Conjecture}$ We may find a regular CW decomposition $(Y_{i})$ of $Y$ such that $f(B_{f}) \subseteq Y_{n-1}$.
The branched covering is a covering on $X \setminus B_{f} \to Y \setminus f(B_{f})$ so we may lift $Y_{i}$ to a regular CW decomposition of $X$?