Curvature, diameter, homotopy groups, and cohomology rings

Xiaochun Rong, Fuquan Fang
2001 Duke mathematical journal  
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 induced
more » ... tions on Y are equivalent. Then {π q (M) finitely generated, M ∈ 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 ≤ and diam M ≤ d, then π q (M) has a finite number of possible isomorphism classes depending on n, λ, , d, q ≥ 2, provided π q (M) is finitely generated. We also show that (B) if M is a compact n-manifold with λ ≤ sec M ≤ and diam(M) ≤ d, then the cohomology ring, H *
doi:10.1215/s0012-7094-01-10717-5 fatcat:tp6jm237snb2nglstpkwxczlwa