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Differential forms are useful for integration in manifolds. But we have already seen that for the integration to be defined rigorously we have to stick to compact support differential forms. Surprisingly enough, this restriction leads to another type of cohomology

The restriction of the de Rham complex to compact support differential forms leads to a new complex

$$ \Omega^0_c(M)\xrightarrow{\mathrm{d}^0}\Omega^1_c(M)\xrightarrow{\mathrm{d}^1} \Omega^2_c(M)\xrightarrow{\mathrm{d}^2}...\xrightarrow{\mathrm{d}^{k-1}}\Omega^k_c(M)\xrightarrow{\mathrm{d}^k}...\xrightarrow{\mathrm{d}^{n-1}}\Omega^n_c(M) $$

and to the cohomology groups with compact support

$$H^k_c(M)=\dfrac{\text{ker }\mathrm{d}^k}{\text{im }\mathrm{d}^{k-1}}=\dfrac{Z^k_c(M)}{B^k_c(M)}$$

If $M$ is a compact, all forms have compact support, $\Omega^k_c(M)=\Omega^k(M)$, and both cohomology groups coincide