I have been trying to figure out if there is an analogous theory using Cobordism to simplicial homology. That is we consider closed submanifolds of a space X directly, as one considers simplices of a simplicial complex directly in simplicial homology. Looking into this, the biggest problem seems to be the operation on the "group", since we can't use a disjoint union of the submanifolds, and their union may not be a closed.
I thought to fix this problem we could consider submanifolds up to homotopy, so that if their intersection mod 2 number was 0 we would be fine. Then this could work over manifolds whose submanifolds always have intersection number 0.
Does this theory already exist? Is the above stated property sufficient? I am having a bit of an issue proving that it is an equivalence relation under only that assumption. If the above might work, what kind of manifolds have this property if any? It seems like $\mathbb{R}^n$ and maybe spheres, at least $S^2$. I wonder if this is true for S^n?
EDIT: Mod 2 intersection theory may not work here, what we truly need is a manifold M with the property that ever pair of k-dimensional submanifolds may be homotoped apart over M.