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Suppose $\alpha$ is a degree $k$ complex valued smooth form on a smooth manifold. The integration of $\alpha$ along integral homology cycles gives its periods. Is it correct that the periods are discrete subgroups of $\mathbb{C}$.

Thanks the set of periods a proper subgroup of $\mathbb{C}$.

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No. Consider for example the torus $T^2 =S^1\times S^1$ with angular coordinates $\theta_1,\theta_2\in\mathbb{R}\bmod 2\pi\mathbb{Z}$ and the closed $1$-form

$$\alpha =\frac{1}{2\pi}\bigl(\; d\theta_1 +\sqrt{2} d\theta_2\;\bigr).$$Its periods are

$$\bigl\{ \; n_1+n_2\sqrt{2};\;\;n_1,n_2\in \mathbb{Z}\;\bigr\}, $$

and they form a dense subset of $\mathbb{R}$.

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To put this in perspective: Every homomorphism from the integral homology group $H_k(M)$ to $\mathbb C$ arises from some closed $k$-form on $M$ (and the form is unique modulo exact forms). This is the content of the de Rham isomorphism. – Tom Goodwillie Nov 22 '13 at 14:06

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