Curvature, diameter, homotopy groups, and cohomology rings

Fuquan Fang, Xiaochun Rong

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

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 πq(M), MNG(πq (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 {πq(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(πq(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≤ secM≤Λ and diam M≤d, then πq(M) has a finite number of possible isomorphism classes depending on n, λ, Λ, d, q≥2, provided πq(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].

Original language English (US) 135-158 24 Duke Mathematical Journal 107 1 https://doi.org/10.1215/S0012-7094-01-10717-5 Published - 2001

All Science Journal Classification (ASJC) codes

• Mathematics(all)

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