It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod $2$ homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.
"Realizing homology classes up to cobordism." Osaka J. Math. 54 (4) 801 - 805, October 2017.