Although Vietoris–Rips (VR) complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes show an interesting progression of homotopy types as the scale increases, moving from the circle S^1 to S^3 to S^5, and so on. However, much less is known about VR complexes of higher-dimensional spheres.

I will present work exploring the VR complexes of the n-sphere S^n and show how the appearance of nontrivial homotopy groups of these complexes can be controlled by the covering properties of S^n and real projective space RP^n. Specifically, if the first nontrivial homotopy group of a VR complex of S^n at scale π - t occurs in dimension k, then S^n can be covered by 2k + 2 balls of radius t, but there is no covering of RP^n by k balls of radius t/2. This is joint work with Henry Adams and Žiga Virk.