Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry

Abstract.

In this paper, we study complete Riemannian n-manifolds (n ≥ 3) with asymptotically nonnegative Ricci curvature and weak bounded geometry. We show among other things that the total Betti number of such a manifold has polynomial growth of degree n ² + n. Further more, such a manifold is of finite topological type if the volume growth rate of the metric ball around the base point is less than $1+ \frac{1}{n}.$