Curvature, Diameter and Bounded Betti Numbers

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.

Keywords

Diameter / Ricci curvature / Sectional curvature / Bounded cohomology / Bounded Betti number / 53C21 / 53C23

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