Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0 < δ ≤ sec_M ≤ 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)δ = frac(1, 4 (1 + 10^{- 6})²) and the fundamental group is a non-cyclic group of order ≥C, a constant;(iii)The volume is less than ε{lunate} (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.