Realizing homology classes up to cobordism
  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. 
Mark Grant, Andr\'as Sz\H{u}cs, Tam\'as Terpai
math.AT