Pyramid Ricci flow in higher dimensions

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold ( M n , g 0 ) that is PIC1, or more generally satisfies a lower curvature bound K IC 1 ≥ - α 0 . That is, instead of constructing a flow on M × [ 0 , T ] , we construct it on a subset of space-time that is a union of parabolic cylinders B g 0 ( x 0 , k ) × [ 0 , T k ] for each k ∈ N , where T k ↓ 0 , and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed IC 1 -limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.