Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness
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Submission date: 01. Sep. 1999 (revised version: September 2001)
published in: Mathematische Annalen, 322 (2002) 2, p. 413-420
DOI number (of the published article): 10.1007/s002080100281
MSC-Numbers: 53C20, 53C21, 53C23, 57N99, 57R57
Keywords and phrases: diffeomorphism finiteness, homotopy group finiteness, nonnegative curvature, positive curvature
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Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 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. Thus in these dimensions the lower positive bound on volume in Cheeger's Finiteness Theorem can be replaced by a purely topological condition, simply-connectedness. In dimension 4 instead of simply-connectedness here only non-vanishing of the Euler characteristic has to be required.
As a topological corollary we obtain that for k+l<7 there are over a given smooth closed l-manifold only finitely many principal Tk bundles with simply connected and non-diffeomorphic total spaces.
Furthermore, for any given numbers C and D and any dimension m it is shown that for each i N there are up to isomorphism always only finitely many possibilities for the ith homotopy group of a simply connected closed m-manifold which admits a metric with curvature |K| <= C and diameter <=D.