Duke Mathematical Journal

Curvature, diameter, homotopy groups, and cohomology rings

Fuquan Fang and Xiaochun Rong

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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 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 $\mathscr {M}$(n,Y) be the collection of compact orientable n-manifolds whose oriented bundles admit SO(n)-invariant fibrations over Y with fiber compact nilpotent manifolds such that the induced SO(n)-actions on Y are equivalent. Then {πq(M) finitely generated, M∈$\mathscr {M}$(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 diamMd, 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 λ≤ secM≤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].

Article information

Duke Math. J., Volume 107, Number 1 (2001), 135-158.

First available in Project Euclid: 5 August 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 55P62: Rational homotopy theory 55Q05: Homotopy groups, general; sets of homotopy classes


Fang, Fuquan; Rong, Xiaochun. Curvature, diameter, homotopy groups, and cohomology rings. Duke Math. J. 107 (2001), no. 1, 135--158. doi:10.1215/S0012-7094-01-10717-5. https://projecteuclid.org/euclid.dmj/1091736139

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