Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set $$ T_{d \alpha}(\beta) := \int_M \beta \wedge d \alpha \ .$$
The question is : does $T_{d \alpha}$ belong to the closure of the vectorial space generated by integration currents over closed homologically trivial smooth submanifolds of dimension $d-p$ ?
More precisely, does there exist a family $(\Sigma_n)_{n \in \mathbb{N}}$ of embedded closed homologically trivial submanifolds of dimension $d-p$ of $M$ such that
$$ \frac{1}{N} \underset{ 0 \le n \le N}{\sum} \int_{\Sigma_n} \underset{N \to \infty}{\longrightarrow} T_{d \alpha} \ , $$
where $\int_{\Sigma_n}$ stands for the integral current over $\Sigma_n$ and where the convergence is in the sense of the flat norm ? After a quick look at the literature, it looks appears to me as a natural topology in this context, but I would be interested in any other non trivial kind of convergence (whenever it guarantees convergence of masses).
My question is supported by the fact that it seems to me that the co-area formula shows that the above statement holds for the differential $df$ of a function $f$ : in this case I have the impression that one can consider for $\Sigma_n$ a countable and dense (in $df \neq 0$) family of regular level sets of the function $f$.
