Geometric construction of bases of $H_2(\bar{\mathrm{\Omega }},\mathrm{\partial }\mathrm{\Omega },\mathbb{Z})$

We present an efficient algorithm for the construction of a basis of $H_2(\bar{\mathrm{\Omega }},\mathrm{\partial }\mathrm{\Omega };\mathbb{Z})$ via the Poincar'e–Lefschetz duality theorem. Denoting by $g$ the first Betti number of $\bar{\mathrm{\Omega }}$ the idea is to find, first $g$ different $1$-boundaries of $\bar{\mathrm{\Omega }}$ with supports contained in $\mathrm{\partial }\mathrm{\Omega }$ whose homology classes in $\mathbb R^3 \setminus \Omega$formabasisof$H_1(\mathbb R^3 \setminus \Omega;\mathbb Z)$,andthentoconstructin$\overline{\Omega}$ a homological Seifert surface of each one of these $1$-boundaries. The Poincar'e–Lefschetz duality theorem ensures that the relative homology classes of these homological Seifert surfaces in $\bar{\mathrm{\Omega }}$ modulo $\mathrm{\partial }\mathrm{\Omega }$ form a basis of $H_2(\bar{\mathrm{\Omega }},\mathrm{\partial }\mathrm{\Omega };\mathbb{Z})$. We devise a simply procedure for the construction of the required set of $1$-boundaries of $\bar{\mathrm{\Omega }}$ that, combined with a fast algorithm for the construction of homological Seifert surfaces, allows the efficient computation of a basis of $H_2(\bar{\mathrm{\Omega }},\mathrm{\partial }\mathrm{\Omega };\mathbb{Z})$ via this very natural geometrical approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.