Persistent Sullivan Minimal Models of Metric Spaces
Ling Zhou
TL;DR
The paper introduces persistent Sullivan minimal models to import rational homotopy theory into persistence, defining a coherent algebraic framework for persistent spaces and proving a stability result: the interleaving distance in the CDGA homotopy category is bounded by the homotopy interleaving distance of the underlying persistent spaces. For Vietoris–Rips filtrations, this yields invariants that are strictly more discriminative than standard persistent homology and provides sharper lower bounds on the Gromov–Hausdorff distance than approaches based on persistence alone or persistent rational homotopy groups. The work also establishes two robust lower bounds via indecomposables: one from persistent cohomology and another from the linear parts of CDGAs, which together imply that persistent Sullivan models capture more information than cohomology alone. Existence, uniqueness, and functoriality of persistent Sullivan minimal models are developed, with concrete corollaries for VR filtrations of metric spaces and connections to rational homotopy groups. Overall, the framework promises more powerful, algebraically tractable invariants for comparing shape across scales in topological data analysis.
Abstract
We extend classical tools from rational homotopy theory to topological data analysis by introducing persistent Sullivan minimal models of persistent topological spaces. Our main result establishes that the interleaving distance between such models in the homotopy category of CDGAs is stable with respect to the homotopy interleaving distance of the underlying spaces. For Vietoris-Rips filtrations of metric spaces, this yields new persistent invariants that are more discriminative than persistent homology. We further show that these models provide sharper lower bounds for the Gromov-Hausdorff distance than those obtained from persistent homology or persistent rational homotopy groups.