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We get some results on the filtration invariant in the homotopy of spheres. Then by the relation between the dilatation invariant and filtration invariant, we get the corresponding results on the dilatation invariant in the homotopy of spheres.

We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.