Curvature, diameter and bounded Betti numbers

We consider the image of the comparison homomorphism from bounded cohomology to ordinary cohomology. Its dimension is called the bounded Betti number. In our earlier work we proved that the total bounded Betti number of a closed Riemannian $n$-manifold satisfying $\text{Ric}\ge -(n-1)$ and $\text{Diam}\le D$ is bounded by a constant depending only on $n$ and $D$. The original proof used the amenability of the images of sufficiently small local fundamental groups. Since that work was written, Kapovitch and Wilking proved Gromov’s generalized Margulis lemma under a lower Ricci curvature bound. Their theorem gives the stronger conclusion that these local groups are uniformly virtually nilpotent. We explain how their result supplies a direct modern input for the bounded Betti number theorem. We also discuss the topological meaning of the bounded part of ordinary cohomology and its relation to the ${\mathrm{\ell }}^1$-seminorm on homology.