Finiteness of rank invariants of multidimensional persistent\nhomology groups

Abstract

Persistent topology is a theory for studying objects related to computer vision and computer graphics. It involves\nanalyzing the qualitative and quantitative behavior of real-valued functions defined over topological spaces. This is\nachieved by considering the filtration obtained from the sequence of nested lower level sets of the function under study,\nand by encoding the scale at which a topological feature (e.g., a connected component, a tunnel, a void) is created, and\nwhen it is annihilated along this filtration. In this framework, multidimensional persistent homology groups capture the\nhomology of a multi-parameter increasing family of spaces. For application purposes, these groups are further encoded by\nsimply considering their rank, which yields a parameterized version of Betti numbers, called rank invariants .\nIn this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in the euclidean n-dimensional space .