Removal and Contraction Operations in $n$D Generalized Maps for Efficient Homology Computation
This work addresses computational efficiency in topological data analysis, specifically for homology computation in geometric modeling, and appears incremental as it builds on existing operations with new conditions.
The paper tackles the problem of efficiently computing homology generators for nD generalized maps by proving that contraction operations preserve homology under certain conditions, resulting in a significant reduction in the number of cells while maintaining the same homology.
In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D generalized maps. Its principle consists in simplifying a generalized map as much as possible by using removal and contraction operations. We obtain a generalized map having the same homology than the initial one, while the number of cells decreased significantly. Keywords: $n$D Generalized Maps; Cellular Homology; Homology Generators; Contraction and Removal Operations.