A sphere theorem with a pinching constant below \({1\over 4}\) (Q678800)

The authors prove the following sphere theorem: There exists a constant \(\delta_{\text{odd}}\in(0,{1\over 4})\) such that any odd-dimensional, compact, simply connected Riemannian manifold \(M^n\) with \(\delta_{\text{odd}}\)-pinched sectional curvature is homeomorphic to the sphere \(S^n\). It is worth noticing that the constant \(\delta_{\text{odd}}\) is independent of the dimension and explicit. Furthermore, in the even-dimensional case, the authors find a pinching constant \(\delta_{\text{ev}}<{1\over 4}\) which guarantees that the integral cohomology groups of \(M^n\) coincide up to torsion groups of odd order with the cohomology groups of a sphere or a projective space. Both results are reduced, by virtue of the diameter sphere theorem of Grove and Shiohama, to proving Berger's horseshoe conjecture under a suitable condition on the diameter. The authors introduce refined estimates which do not follow from standard Jacobi field estimates. These estimates comprise a new tool in comparison geometry, which may be useful in other contexts.