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I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it.

Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold?

  1. $F_0 = f$;
  2. $F_t : M \to \overline{M}$ is an immersion for every $t \in [0,1]$;
  3. $F_1(M)$ has only transverse self-intersections.

(Here, $F_t(p) = F(p,t)$ for every $(p,t) \in M \times [0,1]$.)

If this does not hold in this full generality, is it true for hypersurfaces ($k=1$)? For $\overline{M} = \mathbb{R}^{n+k}$?

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