  In this paper, we study the topology of complete noncompact Riemannian
manifolds with asymptotically nonnegative Ricci curvature and large volume
growth. We prove that they have finite topological types under some curvature
decay and volume growth conditions. We also generize it to the manifolds with
$k$-th asymptotically nonnegative Ricci curvature by using extensions of
Abresch-Gromoll's excess function estimate.
