Suppose that $M^n \subset \mathbb{R}^N$ is a regular submanifold of $\mathbb{R}^n$ diffeomorphic to $\mathbb{RP}^n$. Calculate $H^{k}_{dr}(\mathbb{R}^n \setminus M^n)$ for each $k \in \mathbb{Z}$.
I thought about using the Jordan separation theorem and then applying Mayer Vietoris but that won't work since $N \neq n+1$. But other than that I don't have any other ideas. I'd be grateful for any help!