Collapsed 5-manifolds with pinched positive sectional curvature

Abstract

Let M be a closed 5-manifold of pinched curvature 0<δ\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ=1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has index at least w(δ), a constant depending on δ. (iii) The ratio of the volume and the maximal injectivity radius is less than ε(δ). (iv) The volume is less than ε(δ) and the fundamental group π_1(M) has a center of index at least w, a universal constant, and π_1(M) is either isomorphic to a spherical 5-space group or has an odd order.