Thom-Pontryagin construction gives the 1-1 correspondence between
framed cobordism classes of $k$-dimensioanl sub-manifolds of $S^{n+k}$
and
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.)