A topological sphere theorem for arbitrary-dimensional manifolds (Q1047589)

The existing sphere theorems (like the ones proved by Coghlan and Itokawa in 1991 and the author in 2004) are related to compact, simply connected Riemannian manifolds of even dimension. If the sectional curvature, the injectivity radius and the volume of such a manifold satisfy some bound conditions, then the Riemannian manifold is homeomorphic to \(S^{n}\). In this article, the author proves a similar sphere theorem for a compact, connected Riemannian manifold, of arbitrary finite dimension, weakening the assumptions on its sectional curvature and injectivity radius.