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MiS Preprint
57/1999
Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness
Wilderich Tuschmann
Abstract
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 $i \in N$ 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.
