Motivation I have learned algebraic topology. In simplical homology, we define $C_k(X)$ as an abelian group freely generated by $k$-dimensional skeleton $X^{(k)}$, and boundary operator $\partial_k$ actually takes the $k-1$ dimensional faces.
So can we do the same on topological (or smooth) manifold? Let $M$ be an $m$-dimensional manifold. $D_k(M)$ consists of $k$-dimensional submanifolds and $\partial_k$ takes their boundary.
Does this concept exist? Is there any reference about this? Any advice is helpful. Thank you.