Collapsed manifolds with pinched positive sectional curvature

Let Mⁿ be a manifold of sectional curvature, 0 < δ < K_Mⁿ≤1, let X be an Alexandrov space of curvature ≥ −1. Suppose the Gromov-Hausdorff distance of Mⁿ and X is less than ∈(n,δ) > 0. Our main results are: (A) If X has the lowest possible dimension, n−1/2, then a covering space of Mⁿ of order ≤ n+1/2 is diffeomorphic to a lens space, Sⁿ/ℤ_q, such that 0 < c(n,δ)[vol(Mⁿ)]⁻¹ ≤ q ≤ vol(S_δⁿ)[vol(Mⁿ)]⁻¹, where S_δⁿ is the sphere of constant curvature δ. (B) If X has nonempty boundary, then a coveringspace of Mⁿ of order ≤ n+1/2 is diffeomorphic to a lens space, provided ∈ depends also on the Hausdorff measure of X.