Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected? It's essentially the same transversality argument one uses to show $\pi_k S^n$ is trivial for $k < n$. Only you notice you can make the map simultaneously transverse to a countable collection of manifolds.

What manifolds are bounded by RP^odd? @sara: I'm using the standard linear actions, so that's why the spaces are standard. Are you wondering whether or not one can construct similar bundles with non-standard smooth structures on two of the three spaces?