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Let $X=T^2-\{p\}$ be the torus with one point removed. Since ${p}$ is closed in $T^2$, $X=T^{2}-\{p\}$ is open. In Hausdorff spaces compact subsets are closed, so $X$ is not compact.

I was wondering how to compute the de Rham cohomology with compact supports of $X$.

It's not the same as the standard de Rham cohomology. As $X$ is orientable and connected, $H^2_c(X)=\mathbb R$, but I am struggling with $H^0_c$ and $H^1_c$.

Paweł Czyż
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NicAG
  • 661

1 Answers1

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Using Poincaré duality $$H_c^k \simeq H_{2-k}$$ you can reduce the computation to this answer.

Paweł Czyż
  • 3,238
  • Since $M=T^2-{p}$ is oriented one could also use the duality $H^k(M)=(H^{n-k}_c(M))^$. So $H^{2}_c(M)=\mathbb{R}^$ , $H^{1}_c(M)={\mathbb{R}^{2}}^{}$ and $H^{0}_c(M)=0^$. Is that right? – NicAG Apr 22 '20 at 17:20
  • Yes, this works as well. – Paweł Czyż Apr 22 '20 at 20:14