Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

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 $\mathbb{R}$.

share|improve this answer
2  
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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