Curvature, Diameter and Bounded Betti Numbers.

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 (Ĥ*(M),ǁ · ǁ∞) among closed Riemannian manifold (M,g) with K(M)≥ -1 and Diam (M)≤D. [ABSTRACT FROM AUTHOR]