This paper deals with the homology computation of a subdivided object during its construction. In this paper, we focus on the construction operation consisting of merging cells. For each step of the construction, a homological equivalence is maintained. This algebraic structure connects the chain complex associated with the object to a smaller object (i.e. containing less cells) having the same homology. So, homology computation is achieved on this smaller object more efficiently than on the constructed object, due to their respective sizes. We prove that, at each step, maintaining the homological equivalence has a complexity depending only on the size of the part of the object impacted by the operation. We define a convenient data structure based on sparse matrices that guarantees this result in practice, and show some experimental results obtained with its implementation. Moreover, the method may also be used to compute homology groups generators of any dimension at the cost of an increased complexity.

本文讨论了细分对象在构造过程中的同源性计算。在本文中，我们专注于由合并单元格组成的构造操作。对于构造的每个步骤，均保持同源性。该代数结构将与对象关联的链复合体连接到具有相同同源性的较小对象（即，包含更少的单元）。因此，由于其较小的大小，因此在此较小的对象上比在构造的对象上更有效地实现同源性计算。我们证明，在每个步骤中，保持同等性的复杂性仅取决于操作所影响的对象部分的大小。我们基于稀疏矩阵定义了一种方便的数据结构，该结构可在实践中保证这一结果，并显示了其实施获得的一些实验结果。此外，该方法还可以用于以增加的复杂性为代价来计算任何维度的同源性组生成器。