A generalized π2-diffeomorphism finiteness theorem

Abstract

The π2-diffeomorphism finiteness result of F. Fang-X. Rong and A. Petrunin-W. Tuschmann (independently) 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 π2-diffeomorphism finiteness by removing the condition that π1(M) = 0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M.