We prove that if a closed Riemannian manifold \((M^n,g)\) has finite fundamental group and satisfies the curvature condition \(\begin{aligned} R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left( R_{1212} + R_{3434}\right) \end{aligned}\) for all orthonormal four-frame \(\{e_1, e_2, e_3, e_4\} \subset T_pM\) , then the universal cover of M is homeomorphic to the n-sphere. This generalizes the famous sphere theorem under the stronger condition of \(\frac{1}{4}\) -pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990.
