On the structure of almost nonnegatively curved manifolds (Q1203617)

It follows from a well known result by Cheeger-Gromoll that a compact Riemannian manifold with nonnegative sectional curvature is, up to a finite cover, diffeomorphic to a direct product of a torus and a simply- connected manifold. In this paper a sufficient condition in terms of the diameter and the volume is given for a compact almost nonnegatively curved manifold to be, up to a finite cover, homeomorphic to the product of a torus and a simply-connected manifold. Techniques from geometric topology about controlled maps, manifold resolutions and thin \(h\)- cobordism theorem are used.