# Amenable covers and l^1-invisibility

## TL;DR

This paper proves that in topological spaces with an amenable cover of finite multiplicity, all real homology classes in degrees at least the cover's multiplicity become trivial in -homology, strengthening earlier results on -seminorms.

## Contribution

It establishes that homology classes in certain spaces vanish in -homology when the space admits an amenable cover of bounded multiplicity, extending previous results on -seminorms.

## Key findings

- Homology classes in degrees  at least the cover's multiplicity vanish in -homology.
- The result generalizes Gromov and Ivanov's findings on -seminorms.
- Provides a new criterion for -invisibility in topological spaces.

## Abstract

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.