TITLE: Manifolds whose boundaries are real / complex projective spaces
QUESTION [6 upvotes]: What are some simple manifolds $M$ whose boundaries are $\mathbb{RP}^2$ or $\mathbb{RP}^n$ in general? 

Namely $\partial M= \mathbb{RP}^n.$

What are some simple manifolds $N$ whose boundaries are $\mathbb{CP}^2$ or $\mathbb{CP}^n$ in general? 

Namely $\partial N= \mathbb{CP}^n.$
Some trial attempt: I know that we can have
$\partial D^2= \mathbb{RP}^1.$ and
$\partial D^3= \mathbb{CP}^1.$ I dont know whether there are simple topology that also do the job for the above two questions?

REPLY [6 votes]: This should really be a comment on Mike's excellent answer, but it won't fit.  As Mike twice uses the fact that "$\chi(M)$ odd implies $M$ is not a boundary", I figure I would add a proof of the fact.
Let $N$ be a manifold with $\partial N = M$.  The goal is to show that $\chi(M)$ is even.  If $N$ is even dimensional, $M$ is an odd dimensional closed manifold, so $\chi(M) = 0$, which is even.  Thus, we may assume that $N$  is odd dimensional.
Now, consider the manifold $N'$ obtained by gluing to copies of $N$ together along the commmon boundary.  Then $\chi(N') = 0$ since $N'$ is an odd dimensional closed manifold.  But we also see that $\chi(N') = \chi(N) + \chi(N) - \chi(M)$, so $ \chi(M) = 2\chi(N)$ is even.