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?