Index vector-function and minimal cycles
The present paper takes into account the group of simplicial 1-chains \(C_1(P)=C_1(P; \mathbb Z_2)\) and the homology group \(H_1(P)=H_1(P; \mathbb Z_2),\) where \(P\) is a closed triangulated manifold, \( dim(P)=n\). Starting from a non-negative weighting function \(L: C(P) \to \mathbb R\) [see \textit{T. H. Cormen, C. E. Leiserson, R. L. Rivest} and \textit{C. Stein}, Introduction to algorithms. 2nd ed. Cambridge, MA: MIT Press (2001; Zbl 1047.68161)] and making use of a simplicial scheme of the space of the regular covering \(p: \hat P \to P\) with automorphism group \(G \equiv H_1(P)\), three algorithms are exposed, with the following aims: (1) construction of the index vector-function relative to the basis of the group \(H_{n-1}(P);\) (2) searching for the minimal path homologous to the given 1-chain; (3) searching for the minimal cycle from a fixed homology class.
scientific article; zbMATH DE number 5141808