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I need a reference for (what I believe is) a theorem.

Suppose I have a closed differential $k$-form $\omega$ on a closed manifold $M^n$. Suppose $H_k(M; \mathbb{R}) \cong \mathbb{R}^q$. Then for each generator $[z_i]$ of $H_k(M; \mathbb{R})$, I can find a closed, orientable $k$-manifold $N^k_i$ which does not bound in $M$, called a carrier of that generator of $H_k(M; \mathbb{R})$, $[z_i]$. We call $\int_{N_i} \omega$ the period of $\omega$ over $[z_i]$.

The theorem (which I believe is due to de Rham himself) purportedly states that $\omega$ is exact if and only if its period over every generator of $H_k(M; \mathbb{R})$ is 0.

Does anyone know of a reference for or a proof of this theorem?

Thanks much in advance.

[I think more is true:

For each generator $[z_i]$ of $H_k(M; \mathbb{R})$, there is a unique generator $[w_i]$ of $H^k_{dR}(M)$ and a closed but not exact form $\eta_i$, again called a carrier of $[w_i]$, with $\int_{N_i} \eta_i \ne 0$.

From this statement, the first one follows, for if $\int_{N_i} \omega = 0$ for all $i$ from 1 to $q$, then $\omega$ must not be a linear combination of any (net) non-zero multiples of any carriers of $H^k_{dR}(M)$, and so, by the definition of $H^k_{dR}(M)$, $\omega$ must be exact.

Again, any help in proving or providing a reference for either statement is appreciated.]

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I guess this is a reference: G. de Rham, "Sur l'analysis situs des variétés à dimensions" J. Math. Pures Appl. Sér. 9, 10 (1931) pp. 115–200

encyclopediaofmath.org/index.php/De_Rham_theorem

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