Pinching and concordance theory
It is known that a complete simply connected manifold \(M\) whose sectional curvature \(sec(M)\) satisfies \(1/4 < sec(M) \leq 1\) is homeomorphic to a sphere (Rauch-Berger-Klingenberg). Beyond that, the diffeomorphism type of \(M\) is subject to the following condition (from ``Theorem A'' and ``Remark 1.8''). Let \(W(M)\) denote the space of smooth Morse functions on \(M\) having exactly two critical points. With the above hypotheses on \(M\), the map from \(W(M)\) to \((M\times M)\setminus \Delta\) (where \(\Delta =\)diagonal) which sends \(f\) to the ordered pair of its critical points has a section respecting the canonical involutions. (The involutions send \(f\) to \(-f\), and \((x,y)\) to \((y,x)\).) Methods from concordance theory and algebraic \(K\)-theory show that many exotic spheres do not satisfy the condition.
scientific article; zbMATH DE number 488228