Geometric construction of bases of $H_2(\overline\Omega, \partial\Omega,   \mathbb{Z})$
  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. 
Ana Alonso Rodr\'iguez and Enrico Bertolazzi and Riccardo Ghiloni and   Ruben Specogna
math.AT