I am trying to calculate $H_k(X)$ where $X = \mathbb{R}P^n – \{ x_0 \}$
I started thinking about $k=2$. We can get the projective plane by taking the upper hemisphere with points along the equator identified according to the antipodal map. If we remove a point from the hemisphere, we can then enlarge this such that we are just left with a circle and thus for $k=2$ we just have the homology of the circle.
My geometric intuition starts to fail for $k=3$ and higher spaces. So my questions are:
1) Does the same construction work for higher $k$? (probably not, this seems too easy)
2) If not, what is the nice way to calculate the homology groups (say we know $H_k(\mathbb{R}P^n)$? I guess there is a way to use Mayer-Vietoris, but I just can't see it
Best Answer
Your idea should work similarly in general: Take the closed upper hemisphere of $S^n$. Then $\mathbb{R}P^n$ is obtained by identifying antipodal points on the equator, which is an $S^{n-1}$. So removing a point (preferably not on the equator!) would result in something homotopy equivalent to an $S^{n-1}$ with its antipodal points identified- and that's just $\mathbb{R}P^{n-1}$. So if you know the homology of that, you're in business!