I would like to know how to tackle questions of the following type:
Show that $\mathbb{CP}^{2n}$ is not the boundary of any manifold.
Another such question would be:
Let $\iota: S^1 \to S^3$ be a smooth embedding and $K$ be the image of $\iota$. Show that there is a compact orientable surface $F$ embedded in $S^3$ with $K$ as boundary.
More generally, I would like to know what algebraic topological (homology/cohomology/homotopy theory) tools exist that say anything about what spaces can exist as boundaries.
A reference to an appropriate section in, say Hatcher or Bredon that deals with similar questions would be completely fine too.