Multiparameter persistent homology via generalized Morse theory
arXiv:2107.08856
Abstract
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce indecomposable persistence modules with arbitrarily large dimension. In the case of smooth functions on a compact manifold, we apply cobordism theory and Cerf theory to study the resulting persistence modules. We give examples in which we obtain a complete description of the persistence module as a direct sum of indecomposable summands and provide a corresponding visualization.
21 pages, added proof of stability, added statistical motivation, improvements in exposition thanks to referee's comments