Tagged Barcodes for the Topological Analysis of Gradient-Like Vector Fields

Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals (briefly, a tagged barcode). We show how to apply this pipeline to the weighted and based Morse chain complex of a gradient-like Morse–Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings. Interestingly for computations, it turns out that there is an isometry between factored chain complexes endowed with the interleaving distance and their tagged barcodes endowed with the bottleneck distance. Concerning stability, we show that the map taking a generic enough gradient-like vector field to its barcode of tagged intervals is continuous. Finally, we prove that the tagged barcode of any such vector field can be approximated by the tagged barcode of a combinatorial version of it with arbitrary precision.