Curvature, diameter and bounded Betti numbers
Abstract
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 Ric≥ −(n−1) and Diam≤ 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 ℓ1-seminorm on homology.
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BibTeX
@misc{shen2026curvature,
title = {{Curvature, diameter and bounded Betti numbers}},
author = {Zhongmin Shen and Jyh-Yang Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18530},
eprint = {2610.18530},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18530},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature, diameter and bounded Betti numbers
AU - Zhongmin Shen
AU - Jyh-Yang Wu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18530
UR - https://arxiv.org/abs/2610.18530
AB - 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 Ric≥ −(n−1) and Diam≤ 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 ℓ^1-seminorm on homology.
ER -