The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms.

中文翻译：

---

有限拓扑空间上的有效同调计算

有限拓扑空间的拓扑不变量的研究是相关的，因为它们可以用作多种拓扑空间的模型，包括规则的 CW 复形。在这项工作中，我们为 Kenzo 系统提供了一个新模块，该模块允许在不同情况下使用有限拓扑空间生成器计算同调群。我们的算法将有关拓扑空间的著名结果的新构造版本与用于有限空间的组合方法相结合。在 h 正则空间的特殊情况下，实现了有效且合理高效的方法，并应用了离散向量场技术以改进先前的算法。