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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.