Efficient construction of 2-chains representing a basis of $H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})$
Tóm tắt
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.
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