Index Function and Minimal Cycles

Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method proposed in article builds a cycle z with minimal weight L(z). The main idea is in using a simplicial scheme of space of the regular covering with automorphism group H_1(P). We construct this covering applying index function relative to any basis of group H_{n-1}(P).