TITLE: What manifolds are bounded by RP^odd?
QUESTION [38 upvotes]: Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that odd dimensional ones are boundaries of compact $(n+1)$-manifolds. My question is: are there any especially nice constructions of these $(n+1)$-manifolds? 
I'm especially interested in the case $n=3$. I believe we can get an explicit example of a 4-manifold bound by $\mathbb{R}P^3$ using Rokhlin/Lickorish-Wallace but it doesn't look like that would generalize to higher dimensions at all easily. Are there a lot of different 4-manifolds with this property?

REPLY [7 votes]: Here is a simple general fact: if a manifold $M$ admits a free involution $T$, then $M$ is a boundary.  Proof: $M$ will be the boundary of $W =(M \times [-1,1])/((Tx,t) \sim (x,-t))$.  In your particular case: when $n$ is odd, the standard free action of $C_4$ on $S^n$ induces a free action of $C_2$ on $\mathbb RP^n$.  (Note that this $W$ will tend to be nonorientable, even if $M$ is.  Exercise: let $M = S^1$.)