Geometric construction of bases of $H_2(\overlineΩ, \partialΩ, \mathbb{Z})$

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

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