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Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that RicM > 1, KM '-K 2 and the volume of M > the volume of the (r e)-ball on the unit m-sphere, m = dim M. Then for a suitable choice of e = e(m, K) every M belonging to this class is homeomorphic to

A sphere theorem for manifolds of positive Ricci curvature