Original sentence: Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is.significantly easier to study and compute
Edited sentence: A rather glorious failure in fact, since Poincaré.An -dimensional manifold  has a fundamental homology class  (with coefficients in  in general, and in  in the oriented case), defining a natural transformation.Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute.In general, a.See Cobordism as an extraordinary cohomology theory for the relationship between bordism and homology.closed manifold need not be a boundary: cobordism theory is the study of the difference between closed manifolds and those which are boundaries.in general.simultaneously defined both homology and cobordism -- it is just that they are not the same,.which is not an isomorphism in general