  Let $X$ be a topological space admitting an amenable cover of multiplicity
$k\in\mathbb{N}$. We show that, for every $n\geq k$ and every $\alpha\in
H_n(X;\mathbb{R})$, the image of $\alpha$ in the $\ell^1$-homology module
$H_n^{\ell^1}(X;\mathbb{R})$ vanishes. This strenghtens previous results by
Gromov and Ivanov, who proved, under the same assumptions, that the
$\ell^1$-seminorm of $\alpha$ vanishes.
