Index Vector-Function and Minimal Cycles

(Lobachevskii Journal of Mathematics, Vol.22, pp.35-46 )

Let P be a closed triangulated manifold, dimP=n. We consider the group of simplicial 1-chains C₁(P)= C₁(P,Z₂) and the homology group H₁(P)= H₁(P,Z₂). We also use some nonnegative weighting function L:C₁(P)→R. For any homological class [x] ∈ H₁(P) the method proposed in article builds a cycle z∈[x] with minimal weight L(z). The main idea is in using a simplicial scheme of space of the regular covering p: P' → P with automorphism group G ≅ H₁(P). We construct this covering applying the index vector-function J:C₁(P)→Z₂^r relative to any basis of group H_n-1(P), r= rank H_n-1(P).

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