Efficient construction of 2-chains representing a basis of \(H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})\) (Q1633005)

Let $\Omega$ be a connected open subset of ${\mathbb R}^3$ whose closure $\overline\Omega$ is a polyhedron with locally flat boundary $\partial\Omega$, and let ${\mathcal T}$ be a triangulation of $\overline\Omega$. The authors present an algorithm for constructing a collection of 1--cycles from ${\mathcal T}$ that lie within $\partial\Omega$ and whose classes in the first integral homology group $H_1(\overline\Omega)$ are trivial, and whose classes in $H_1({\mathbb R}^3\setminus\Omega)$ form a basis. This result extends the algorithm of \textit{R. Hiptmair} and \textit{J. Ostrowski} [SIAM J. Comput. 31, No. 5, 1405--1423 (2002; Zbl 1001.05046)] to include the case when $\partial\Omega$ is not connected. Moreover, this result can be combined with the authors' previous work on constructing homological Seifert surfaces [\textit{A. A. Rodríguez} et al., SIAM J. Numer. Anal. 55, No. 3, 1159--1187 (2017; Zbl 1385.55001)] to yield an algorithm for computing generators for the relative homology group $H_2(\overline\Omega,\partial\Omega)$ with respect to ${\mathcal T}$. The results of numerical experiments are presented that indicate the efficacy of the algorithm.