If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth **sphere**-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$.

My question is: which framed submanifolds are induced by $\mathbb{R}^n$-valued maps? In other words, what is the condition on a framed submanifold $N$ of codimension $n$ to be the preimage $f^{-1}(0)$ for some $f: M\to\mathbb{R}^n$ transversal to $0$? "Framed-null-cobordand-ness" is probably necessary but not sufficient.

My motivation comes from this slightly more specific question that I'm trying to solve: if $M$ has a boundary and $g: \partial M\to S^n$ is given, then I want to find a common invariant of $f^{-1}(0)$ for all possible extensions $f: M\to D_n$ of $g$ ($D_n$ is the $n$-disc). At first, I though that a full invariant is a framed cobordism class in $M\setminus\partial M$ and that such cobordism classes are in 1-1 correspondence to homotopy classes $[\partial M, S^n]$. If I could prove that a framed cobordism $W$ between $N=f_1^{-1}(0)$ and $N'$ is the zero set of some $F: M\times [0,1]\to \mathbb{R}^n$ ($F$ nonzero on $\partial M)$, then I could use $F$ to define a homotopy between $F_0: (M,\partial M)\to (D_n, S^{n-1})$ and $F_1$ and then adjust it near the boundary so that $F$ is constant on $\partial M$. But I can't prove that $F$ exists.

I also tried to use a quotient $q: D_n/S^{n-1}\to S^n$, find a homotopy to $S^n$ and lift it: however, the homotopy lifting property works only for Serre fibration, which $q$ is not..

Any help will be much appreciated! (If necessary, I can also write more about the broader motivation behind these problems)

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