I want to show that if $w$ is a $p$-form such that its induced cochain on $p$-chains:


$w(\gamma)= \int\_{\gamma} w \in S$


takes values in a discrete set $S \subset \mathbb{R}$ then $w$ must be zero.


* My idea is to say that we know some chains (i.e. the zero chain) will integrate to zero and that there is a way to continuously vary the chain so that the value $w(\gamma)$ should vary continuously as well; hitting values outside of $S$.


Any suggestions or ideas are greatly appreciated!


Thanks, 
CM



A p-form taking discrete values on p-chains must be 0.
You can determine the value of the $p$-form at a point as the limit if integrals over very small $p$-simplices and rescaling. If the integral takes values in a discrete subgroup of $\mathbb{R}$, then you get zero.


