Curvature, diameter, homotopy groups, and cohomology rings

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 bundles admit SO(n)-invariant fibrations over K with fiber compact nilpotent manifolds such that the induced SO (n)-actions on Y are equivalent. Then {π_q(M) finitely generated, M \in ℳ(n, Y)} contains only finite isomorphism classes depending only on n, Y,q. Together with the results of [CG] and [Gr1], from (A) we conclude that (i) if M is a complete n-manifold of nonnegative curvature, then MNG(π_q(M)) is bounded above by a number depending only on n and q≥2. Together with the results of [Ch] and [CFG], from (C) we conclude that (ii) if M is a compact n-manifold whose sectional curvature and diameter satisfy λ≤ sec_M≤Λ and diam _M≤d, then π_q(M) has a finite number of possible isomorphism classes depending on n, λ, Λ, d, q≥2, provided π_q(M) is finitely generated. We also show that (B) if M is a compact n-manifold with Sλ≤ sec _M≤Λ and diam(M)≤d, then the cohomology ring, H* (M,Q), may have infinitely many isomorphism classes. In particular, (B) answers some questions raised by K. Grove [Gro].