Suppose I have a finite CW complex $X$ with $p$-skeleton $X^{(p)}$. 


* Let $\varphi\_f \colon S^p \to X^{(p)}$ be part of the attaching map of a $(p+1)$-cell $f$.
* Let $\Phi\_e \colon D^p \to X^{(p)}$ be part of the attaching map of a $p$-cell $e$ and let $q\_e \colon X^{(p)} \to S^p$ be the map that collapses everything outside of $\Phi\_e(Int(D^p)) \subset X^{(p)}$ to a point.


Suppose that the map $\theta = q\_e \circ \varphi\_f \colon S^p \to X^{(p)} \to S^p$ is surjective. Let $x\_0 = \Phi\_e(0) \in X^{(p)}$. 



> 
> Is it possible to change $\varphi\_f$ by a homotopy to a map $\hat{\varphi}\_f$, such that $\hat{\varphi}\_f^{-1}(x\_0)$ is a finite set of points?
> 
> 
> 


Since $\theta$ is a continuous map between smooth manifolds, it is homotopic to a smooth map. Hence, I can arrange that $\theta^{-1}(x\_0)$ is a finite set of points after a homotopy. But the result of the homotopy may not lift to a map $S^p \to X^{(p)}$. 


I tried to change $\varphi\_f$ only inside the preimage of the open cell in $X^{(p)}$ and keep the rest fixed, but I did not manage to ensure that there are no new points created in the process that are mapped to $x\_0$.



attaching maps in CW complexes
This is essentially Lemma 4.10 of Hatcher's book, which is the key step in proving the cellular approximation theorem. It shows that you can homotope $\varphi\_f$ so that there is an open set $U$ in your $p$-cell $e$ such that $\varphi\_f$ is piecewise linear on the inverse image of $U$. Then a general element of $U$ has only finitely many preimages.


