Amenable covers and l^1-invisibility

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