compactly supported cohomology ring of punctured complex projective spaces.

Question

Is it true that $$H^{*}_{c}(\mathbb{C}\mathbb{P}^{m}\setminus{\{pt\}},\mathbb{Q})=\widetilde{H^{*}}(\mathbb{C}\mathbb{P}^{m}),\mathbb{Q})$$ where $H^{*}_{c}(\mathbb{C}\mathbb{P}^{m}\setminus{\{pt\}},\mathbb{Q})$ and $\widetilde{H^{*}}(\mathbb{C}\mathbb{P}^{m}),\mathbb{Q})$ are compactly supported and reduced cohomology rings.

Answer 1

Yes. In general, if $X$ is non-compact, then $H^*_c(X) \cong \widetilde{H}^*(X^+)$ where $X^+$ denotes the one-point compactification of $X$ (see the second remark at the beginning of page $2$ of these notes). If $X = M\setminus\{pt\}$ where $M$ is compact manifold, then $X^+ = M$ so we see that $H^*_c(M\setminus\{pt\}) \cong \widetilde{H}^*(M)$.