Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ⩽secM⩽1. We prove that M is homeomorphic to a spherical space form if one of the following conditions holds:(i)The center of the fundamental group has index ⩾w(δ), a constant depending on δ;(ii) and the fundamental group is a non-cyclic group of order ⩾C, a constant;(iii)The volume is less than ϵ(δ) and the fundamental group is either isomorphic to a spherical 5-space group or has an odd order, and it has a center of index ⩾w, a constant.