Let $M$ be a smooth manifold of dimension $n$. Let $\alpha \in \pi_i (M)$, for some $i \leq n$, and $f:S^i \to M$ be a map of the sphere representing the homotopy class $\alpha$. Can $f$ always be chosen to be an immersion? I.e. does every homotopy class of the maps from the sphere $S^i$ to $M$ contain an immersion?

This feels like it should be a classical result but I don't know much about this, nor do I have any idea how to approach such a problem. If it is true, could I get a reference to the result, or if it's simple, a hint for the solution? Or is there some subtlety that I'm missing that makes this statement false?

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    $\begingroup$ Certainly not in general when $i=n$. Then an immersion is covering map. I don't know what happens for, say $i=n-1$. $\endgroup$ Nov 6, 2020 at 2:11
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    $\begingroup$ On the other side, for sufficiently large codimension ($i\le\lfloor n/2\rfloor$ I believe) such an immersion exists by approximation/transversality arguments. $\endgroup$
    – Kajelad
    Nov 6, 2020 at 2:27
  • $\begingroup$ Asked and answered on MO: mathoverflow.net/questions/375919/… $\endgroup$ Jan 7, 2021 at 10:31


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