Computing de Rham Cohomology

I'm stuck on the following problem.

Let $$X=S^{n}\setminus A$$, where $$A$$ is the union of $$k\geq 1$$ disks $$D_{k}$$. Use the Mayer-Vietoris sequence to compute the de Rham cohomology $$H_{\mathrm{dR}}^{*}(X)$$.

For $$k=1$$, I see that $$S^{n}\setminus D_{1}$$ is diffeomorphic to $$\mathbb{R}^{n}$$. Since $$\mathbb{R}^{n}$$ is connected, we know that $$H_{\mathrm{dR}}^{0}(\mathbb{R}^{n})\cong\mathbb{R}$$, so $$H_{\mathrm{dR}}^{0}(S^{n}\setminus D_{1})\cong\mathbb{R}$$. However, I have no idea of show to handle the general case (i.e. $$n\geq 0$$ and $$k> 1$$) to find $$H^{n}_{\mathrm{dR}}(S^{n}\setminus A)$$. I know that I need to decompose the spaces to apply Mayer-Vietoris, but I can't seem to figure out anything past this. Any help is appreciated.

• Hint: instead of trying to decompose $X$, try to decompose $S^n$ (for which the homology is known) into two pieces, one of which is $X$. – hunter Nov 26 '18 at 0:56
• @hunter I hadn't thought of it from that perspective. So would I decompose $S^{n}$ as $X=S^{n}\setminus A$ and $A$? – user608571 Nov 26 '18 at 1:01
• yes. You have to thicken them a bit so there's an overlap. – hunter Nov 26 '18 at 1:36

1 Answer

Hint: You have the $$k=1$$ case. By the same reasoning, for $$k>1$$ $$X$$ is diffeomorphic to $$\mathbb R^n$$ with $$k-1$$ disks removed. Use Mayer-Vietoris and induction on $$k$$ to compute the cohomology of such spaces.