We show that almost nonnegatively curved m-manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C.m/-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.
Abstract. We show that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n -manifolds with 0 ≤ sec ≤ 1 , positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec ≥ 0 have infinitely many connected components.
Our main results can be stated as follows:1. For any given numbers m, C and D, the class of m-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature bounded in absolute value by |K| ≤ C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. 2. Given any m and any δ > 0, there exists a positive constant i 0 = i 0 (m, δ) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ric ≥ δ, K ≤ 1, is bounded from below by i 0 (m, δ). In an appendix we discuss Riemannian megafolds, a generalized notion of Riemannian manifolds, and their use (and usefulness) in collapsing with bounded curvature.
IntroductionThis note is a continuation of the work begun in [PetrRTu] (this issue) and centers around the problems of establishing finiteness theorems and injectivity radius estimates for certain classes of closed Riemannian manifolds. Our first result can be stated as follows:Theorem 0.1 (π 2 -Finiteness theorem). For given m, C and D, there is only a finite number of diffeomorphism types of simply connected closed mdimensional manifolds M with finite second homotopy groups which admit
We show that in each dimension 4n + 3, n ≥ 1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.
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