In this post, I’ll discuss a result- of which I’ll prove half- that will force me to go back and write at least one (hopefully many) post(s) on characteristic classes proper. The result is as follows:
Let be a closed, smooth -manifold. Then, there exists a smooth, compact -manifold with boundary if and only if all the Stiefel-Whitney classes of are zero.
Let be the relative fundamental class of the pair . Then, writing for the homological differential, we have .
Now, the tangent bundle is well-defined even up to the boundary, and , where is the trivial rank-one bundle giving the outward normal to . Thus, . It follows that any polynomial in the Stiefel-Whitney classes of is in the image of , so . Thus, as was sought.
The converse is more difficult. As a corollary, note that are (unoriented) cobordant manifolds if and only if all of their Stiefel-Whitney classes concur.