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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.

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The simplest such example is probably the cycle of a line in the real projective plane. It isn't a boundary itself (informally speaking since the plane is projective, there is no "upper half-plane" of which it is the boundary). However twice this cycle is a boundary. One way to see this is to orient the complement of the line (this is just the affine plane, which can be oriented) and then triangulate the plane so that the line is a subcomplex. Then take the sum of all simplices with the orientation induced by the orientation on the open piece. Taking the boundary we get twice the line. –  algori Dec 12 '10 at 6:25
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More generally (but following algori's lead), lens spaces give you plenty of examples $H_1(L_{p,q}) \simeq \mathbb Z_p$. The torsion linking form on lens spaces gives you the homotopy classification so the bounding chains are useful in this instance. –  Ryan Budney Dec 12 '10 at 7:09

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