Local computation of homology variations over a construction process

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.