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.]


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



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.