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I am doing a problem where I am stuck at this point.

Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n>2m, ~\dim X < n-m$ and $\pi_m(M-X)=0$. Then is it true that $\pi_m(M)=0$?

Here $\dim X$ is the topological dimension of the space $X$. If $X$ is a submanifold, then this result is true (I asked the same question on math stack). I want if it is true for a general (closed) subset $X$.

Any hint/reference will be appreciated.

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  • $\begingroup$ Let $f:S_m\to M$ be a continuous map of the sphere. Your condition on dimensions implies that $f$ can be perturbed a little so that the image does not hit $X$. Since this perturbed $f$ is homotopic to a point in $M\backslash X$, $f$ is homotopic to a point in $M$. $\endgroup$
    – Alexandre Eremenko
    Commented Apr 2, 2023 at 12:34
  • $\begingroup$ Smooth out with the Whitney's smoothing and then apply Sard's theorem. $\endgroup$
    – Ben McKay
    Commented Apr 2, 2023 at 13:10
  • $\begingroup$ I assume that when you say you want a general closed subset to have dimension less than $n-m$, that you mean $X$ is covered by a countable union of submanifolds of dimension $n-m-1$, so you can apply Sard. $\endgroup$
    – Ben McKay
    Commented Apr 2, 2023 at 13:11
  • $\begingroup$ @AlexandreEremenko, I do not know if you can perturb $f$ as explained. If $X$ is a submanifold, then we can (by using the transversality argument). Can you elaborate a little more? $\endgroup$
    – Sachchidanand Prasad
    Commented Apr 2, 2023 at 13:26
  • $\begingroup$ @BenMcKay Yes you are right about the dimension less than $n-m$, but how are you applying Sard's to conclude this? $\endgroup$
    – Sachchidanand Prasad
    Commented Apr 2, 2023 at 13:27

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