Abstract
In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups \( {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{H}^{*} {\left( M \right)},{\left\| \cdot \right\|}_{\infty } } \right)} \) among closed Riemannian manifold (M, g) with K(M) ≥ - 1 and Diam (M) ≤ D.
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(Dedicated to the memory of Shiing-Shen Chern)
* Partially supported by a Taiwan NSC grant.
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Shen, Z., Wu*, JY. Curvature, Diameter and Bounded Betti Numbers. Chin. Ann. Math. Ser. B 27, 143–152 (2006). https://doi.org/10.1007/s11401-005-0398-z
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DOI: https://doi.org/10.1007/s11401-005-0398-z