The flipping principle for boundedly acyclic covers and colorings

Based on the theory of multicomplexes and other ideas by Gromov, we describe the flipping principle. This is an easy geometric criterion that forces the vanishing of the Gromov norm of homology classes. We use it to provide proofs based on a common geometric principle of the Vanishing Theorem and the Amenable Reduction Lemma by Gromov in the case of amenable and boundedly acyclic covers and colorings. We also strengthen the conclusion of the Amenable Reduction Lemma by showing that it implies the $\ell^1$-invisibility of the homology classes and not only the vanishing of their Gromov norm.