Function-Rips complexes in persistent homotopy theory: Stability and persistent Latschev theorems

Classical results of Hausmann and Latschev show that Vietoris-Rips complexes can recover the homotopy type of a manifold, even from finite metric spaces that are nearby in Gromov-Hausdorff distance. We prove persistent homotopical versions of these theorems for metric spaces equipped with filtration functions. The central object of study is the so-called persistent homotopy type of the function-Rips complex, a filtered simplicial complex that combines a fixed Rips scale with the filtration data on the underlying space. Using techniques from CAT($\kappa$)-geometry and persistent simplicial homotopy theory, we generalize Latschev’s and Hausmann’s theorems to the setting of spaces with filtration functions and homotopical interleavings. A fundamental ingredient is a new homotopical stability theorem. The fixed-scale function-Rips construction is known not to be globally stable with respect to function Gromov-Hausdorff distance and homotopical interleaving distance. Here, we show that it is nevertheless stable for appropriate choices of the Rips parameter at such pairs $(M,f)$ for which $M$ is a complete metric space of curvature bounded above, and $f$ is a Lipschitz continuous multivariate function.