Is there a generalization of the Thom-Pontryagin construction in the following sense? Let $M$ be a smooth manifold, $\partial M=A\cup B$ where $A$ and $B$ are $m-1$-manifolds with a common boundary $\partial A=\partial B$. I would like to have a correspondence between $$\pi^n(M,A)=[(M,A), (S^n, *)]$$ and something like "classes of (neat ?) framed submanifolds $N$ of $M$ with $\partial N\subseteq B$", the equivalence given by framed cobordisms disjoint from $\partial A\times [0,1]$".

Intuitively, if $*\in S^n$ is the "south-pole", the correspondence could be given by assigning to $f: M\to S^n$ the preimage of the "north-pole", and to a framed submanifold $N$ a map $f$ that maps $N$ to the north-pole, some neighborhood of $N$ to $S^n\setminus\{*\}$ and the rest to $*$. One problem with this is that $M\times [0,1]$ has corners. Is there a way to deal with it and can I get such 1-1 correspondence?

(I'm not sure whether this question is more suitable here or on math.stackexchange.)


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