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

TL;DR

This paper introduces an efficient geometric algorithm for constructing a basis of the second relative homology group of a 3D domain with boundary, leveraging Poincaré–Lefschetz duality and homological Seifert surfaces.

Contribution

The paper presents a novel, efficient method to compute a basis of $H_2(ar{ ext{Ω}}, ext{∂Ω}; extbf{Z})$ using geometric constructions and duality, improving computational performance.

Findings

Algorithm efficiently computes homology bases.

Numerical experiments demonstrate superior performance.

Method outperforms existing algorithms in speed and accuracy.

Abstract

We present an efficient algorithm for the construction of a basis of $H_2(\overline{\Omega},\partial\Omega;\mathbb Z)$ via the Poincar\'e--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 Poincar\'e--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.