# Cohomology with compact support of $T^2-\{p\}$

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 $$H^1_c$$.

• Worth noting that $T^2 - \{p\}$ is open is only an argument that it is not closed given that $T^2 - \{p\}$ is connected. – Keefer Rowan Apr 21 at 22:01

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

• 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 at 17:20
• Yes, this works as well. – Paweł Czyż Apr 22 at 20:14