So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not cobordant.
As far as I can tell, every pair of one- or two-dimensional closed oriented manifolds are bordant, so such an example would seemingly only occur in dimensions 3 or higher. They reason I think this is that since the torus and the sphere are bordant, we can just "iterate" this cobordism in a suitable way and exhaust all possibilities by the classification theorem of compact oriented surfaces.
I was wondering if anyone knows of an example of a non-cobordant pair of closed oriented manifolds which is relatively easy to appreciate. If possible, an excplicit construction would be preferrable.