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 $x_0$.

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    $\begingroup$ Let me suggest that you transfer this question over to math.stackexchange, which is a more appropriate venue for it. $\endgroup$ – Lee Mosher Jan 10 '17 at 19:01
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    $\begingroup$ @LeeMosher Well, I would, but I don't have a math.stackexchange account and will not create one only for this question. Feel free to close it if you find it too trivial, but I couldn't find anything in the literature that was helpful. It could very well be that I am overlooking something very trivial here and I would be thankful for a hint. At least I can assure you that this is not a homework problem :-). $\endgroup$ – Ulrich Pennig Jan 10 '17 at 19:08
  • $\begingroup$ The short answer is, changing the attaching map by a homotopy does not affect the homotopy type of the CW complex. And so if you have to change it then you have to change it. $\endgroup$ – Lee Mosher Jan 10 '17 at 19:14
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    $\begingroup$ If someone needs a motivation for this: My original problem is related to cellular versions of twisted generalised cohomology theories and involves expressing the degree of the map $\theta$ via local degrees, which should be possible if I can find at least one regular value of the form stated in the question. $\endgroup$ – Ulrich Pennig Jan 10 '17 at 20:47
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    $\begingroup$ I don't see why questions that are easy for specialists necessarily belong on MSE. (Sorry to keep banging the same drum.) Also, some of us have sworn off MSE... $\endgroup$ – Yemon Choi Jan 10 '17 at 22:36

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.

  • $\begingroup$ Thank you, Dan! I think I found a solution in the smooth setting as well. One just has to be a little more careful using the Whitney approximation theorem. $\endgroup$ – Ulrich Pennig Jan 11 '17 at 14:05
  • $\begingroup$ That sounds right - cellular approximation can also be proved using Whitney approximation + Sard's theorem. $\endgroup$ – Dan Petersen Jan 11 '17 at 14:19

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