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 $x_0$.