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Thom-Pontryagin construction gives the 1-1 correspondence between

framed cobordism classes of $k$-dimensioanl sub-manifolds of $S^{n+k}$


homotopy classes of maps from $S^{n+k}$ to $S^n$.

Are there any analogous theorem for the $k$-dimensional sub manifolds with boundary of $S^{n+k}$?

(Note that we have the notion (relative)-cobordism between manifolds with boundary.)

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There is a theory for maps $(M,\partial M)\rightarrow (S^n,x_0)$ for some basepoint $x_0\in S^n$. There is a remark in Milnors "topology from the differential viewpoint" at the end of the chapter that discusses the Pontryagin-Thom construction.

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