A Generalized $\pi_2$-Diffeomorphism Finiteness Theorem

The $\pi_2$-diffeomorphism finiteness result (\cite{FR1,2}, \cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolute value of sectional curvature and diameter of $M$. In this paper, we will generalize this $\pi_2$-diffeomorphism finiteness by removing the condition that $\pi_1(M)=0$ and asserting the diffeomorphism finiteness on the Riemannian universal cover of $M$.