A note on pinching sphere theorem. (Q1426588)

The main result reads: For any \(n\in\mathbb N\) there exists \(\eta\in \mathbb{R}\) depending only on \(n\) and such that any closed, simply connected \((2n)\)-dimensional Riemannian manifold \(M\) with sectional curvature \(K_M\in (0, 1]\) and volume \(V(M)\in [(3/2)V(S^{2n}), ((3/2)+\eta)V(S^{2n})]\) is homeomorphic to the sphere \(S^{2n}\). The author claims that the proof of a stronger result in [\textit{B. Wang}, J. Sichuan Univ., Nat. Sci. Ed. 34, No. 5, 581--587 (1997; Zbl 0907.53028)] contains a gap.