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We establish two topological results. (A) If M is a 1-connected compact n-manifold and q ≥ 2, then the minimal number of generators for the qth homotopy group π q (M), MNG(π q (M)), is bounded above by a number depending only on MNG(H * (M, Z)) and q, where H * (M, Z) is the homology group. (C) Let M (n, Y ) be the collection of compact orientable n-manifolds whose oriented frame bundles admit SO(n)-invariant fibrations over Y with fiber compact nilpotent manifolds such that the induceddoi:10.1215/s0012-7094-01-10717-5 fatcat:tp6jm237snb2nglstpkwxczlwa