When can homology classes be realized as the image of fundamental classes of manifolds? The answer is known, and singular bordism groups provide insight into this problem.
Steenrod's Realization Problem
See also
Bordism Group, ManifoldExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Steenrod's Realization Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteenrodsRealizationProblem.html