Name
Title | ||
|---|---|---|
Efficient construction of 2-chains representing a basis 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 ∂Ω 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 a set of 2-chains in \(\overline {\Omega }\) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in \(\overline {\Omega }\) modulo ∂Ω form a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\). We devise a simple procedure for the construction of the required set of 1-boundaries of \(\overline {\Omega }\) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1007/s10444-018-9588-6 | Advances in Computational Mathematics |
Keywords | Field | DocType |
Relative homology, 2-chains with prescribed boundary, Homological Seifert surfaces, Linking number, Perturbation of simple loops and 1-cycles, 65D17, 68U05, 52B05, 55N99 | Betti number,Combinatorics,Linking number,Modulo,Duality (mathematics),Mathematical analysis,Omega,Relative homology,Overline,Mathematics | Journal |
Volume | Issue | ISSN |
44 | 5 | 1572-9044 |
Citations | PageRank | References |
0 | 0.34 | 5 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Ana Alonso | 1 | 65 | 17.55 |
| Enrico Bertolazzi | 2 | 130 | 15.39 |
| Riccardo Ghiloni | 3 | 8 | 2.26 |
| Ruben Specogna | 4 | 28 | 6.38 |