Small regions, persistent homology: an illustrated companion to the construction

Abstract

This companion explains the construction in Homological obstructions to local contractibility in noncollapsed Ricci limits. Boundary-collapse maps detect independent second-homology classes in the ambient limit. Local metric control and a perturbation argument preserve these classes after deletion of an arbitrary set below the stated Hausdorff-dimension threshold. We describe the regularized circle profile, the compensation of spatial Ricci terms, the flat parameters and positive floors, and the annuli and fixed collars that retain each inserted domain. For the isolated example, localized distance estimates identify a unique product-cone tangent without imposing a topologically conical neighborhood. The figures distinguish geometric models, algebraic estimates, and schematic decompositions.