Suppose X is a smooth manifold with homology groups H_p(X). For example let X be an open subset of Euclidean space.
What would be natural examples of cycles that are "weakly homologous to zero" but that are not "strongly homologous to zero"? That is cycles which are themselves not "boundaries" but for which some integer multiple is a boundary.
In particular, I'm thinking of the de Rham approach to homology/cohomology here where we think of the elements of the homology group as being "smooth singular chains" and the elements of the cohomology groups being differential forms that we "integrate" over the chains.