Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<\delta\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) \delta =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(\delta), a constant depending on \delta. (iii) The ratio of the volume and the maximal injectivity radius is less than \epsilon(\delta). (iv) The volume is less than \epsilon(\delta) and the fundamental group \pi_1(M) has a center of index at least w, a universal constant, and \pi_1(M) is either isomorphic to a spherical 5-space group or has an odd order.