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 tame epimorphic parametrized chain complex, and encodes it as a barcode of tagged intervals. We show how to apply this pipeline to the weighted and based 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 tame epimorphic parametrized chain complexes endowed with the interleaving distance and barcodes of tagged intervals 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 barcode of any such vector field can be approximated by the barcode of a combinatorial version of it with arbitrary precision.

翻译：旨在引入一种场拓扑分析方法，我们提出了一种处理流程：该流程以加权且带基的链复形为输入，生成驯顺满射参数化链复形，并将其编码为带标签区间的条形码。我们展示了如何将该流程应用于光滑和离散两种场景下紧黎曼流形上的梯度类Morse-Smale向量场的加权带基链复形。有趣的是，计算结果表明，在交错距离赋予下的驯顺满射参数化链复形与瓶颈距离赋予下的带标签区间条形码之间存在等距映射。关于稳定性，我们证明了将足够通用的梯度类向量场映射到其带标签区间条形码的映射是连续的。最后，我们证明此类任意向量场的条形码可通过其组合版本的条形码以任意精度逼近。