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In this paper,we prove that a complete n-dimensional Riemannian manifold with nonnegative kth-Ricci curvature, large volume growth has finite topological type provided that limr→∞{(vol[B(p,r)]/ωnrn-αM)rk(n-1)k+1(1-α/2)}≤ε for some constant ε> 0. We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to Rn.