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

Abstract

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

arXiv admin note: text overlap with arXiv:1409.5487