Multiparameter persistent homology via generalized Morse theory

Abstract

We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space. 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.