Real projective spaces ℝPⁿ have ℤ/2 cohomology rings ℤ/2[x]/(xⁿ⁺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 ℝP³ 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?

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?