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Let $M$ be a smooth $n$-dimensional compact manifold with boundary. By the Whitney Immersion Theorem, $M$ can be immersed in $\mathbb{R}^{2n-1}$. I am wondering if there is a relative version of this statement.

In particular, let $U$ be an open neighborhood of $\partial M$. Assume that we have a fixed immersion $\bar f : U \to \mathbb{R}^{2n-1}$. Can we always find an immersion $f : M \to \mathbb{R}^{2n-1}$ such that $f |_{\partial M} = \bar f |_{\partial M}$?

I believe this should work in $\mathbb{R}^{2n}$. Since $\bar f$ is an immersion, $d\bar f$ is a bundle monomorphism $TU \to U \times \mathbb{R}^{2n}$ covering $\bar f$. We can think of this as a section of the bundle over $U$ whose fibers are the Stiefel manifold $V_{n}(\mathbb{R}^{2n})$. Since $\pi_{n-1}\left(V_{n}(\mathbb{R}^{2n})\right) = 0$, we can extend the section from $\partial M$ to $M$. Now we can apply Smale-Hirsch to get an immersion $f : M \to \mathbb{R}^{2n}$, and I believe we can choose to keep $\bar f$ on $\partial M$.

Is there a relative version for $\mathbb{R}^{2n-1}$? If not, is it true for some manifolds $M$?

EDIT: This has now been answered on mathoverflow.

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