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I suspect the following statement is wrong:

Let $X$ be a smooth connected manifold, and $V \subseteq X$ be an embedding submanifold of codimension $\geq d+2$ and $i: X-V \rightarrow X$ the inclusion map. Then the map $i_{*}:\pi_{d}(X-V) \rightarrow \pi_{d}(X)$ is an isomorphism.

However, I cannot find any straightforward counterexample. The statement is known to be true when $d=1$, and I think $i_{*}$ is at least injective for general $d$. However, it seems unlikely that it will be surjective in general.

Edit: As Moishe Kohan pointed out in the remark, the original statement is not true for $d=1$ if the objects are not smooth.

hyyyyy
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  • This statement is false already for $d=0$ (and for $d=1$ as well...). – Moishe Kohan Apr 30 '23 at 14:12
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    https://math.stackexchange.com/questions/1426501/why-are-all-knots-trivial-in-4d/4008370#4008370 – Moishe Kohan Apr 30 '23 at 16:14
  • @MoisheKohan Thank you for pointing it out! I was under the impression that smooth approximation can be made into isotopy and apparantly I was wrong. I added the smooth condition into the problem. – hyyyyy May 01 '23 at 17:40

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