For a closed smooth manifold, we can use the Stiefel-Whitney number of the manifold to decide it is boundary or not.
For a closed topology or PL manifold, how to decide it is a boundary of compact Top/PL manifold?
As far as I know, the Top/PL cobordism ring is a litter complex. So I guess they may not have simple way as in the smooth case to decide which closed Top/PL manifold is boundary. I hope I am wrong.