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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.

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  • $\begingroup$ 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 $X$. $\endgroup$ – hunter Nov 26 '18 at 0:56
  • $\begingroup$ @hunter I hadn't thought of it from that perspective. So would I decompose $S^{n}$ as $X=S^{n}\setminus A$ and $A$? $\endgroup$ – user608571 Nov 26 '18 at 1:01
  • $\begingroup$ yes. You have to thicken them a bit so there's an overlap. $\endgroup$ – hunter Nov 26 '18 at 1:36
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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.

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