# Index Vector-Function and Minimal Cycles

## (*Lobachevskii Journal of Mathematics, Volume XXII*)

Let *P* be a closed triangulated manifold, *dimP=n*.
We consider the group of simplicial 1-chains *C*_{1}(P)=
C_{1}(P,Z_{2}) and the homology group *H*_{1}(P)=
H_{1}(P,Z_{2}). We also use some nonnegative weighting function
*L:C*_{1}(P)→R. For any homological class *[x] ∈ H*_{1}(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 ≅ H*_{1}(P). We construct this covering applying the index
vector-function *J:C*_{1}(P)→Z_{2}^{r} relative to any basis of
group *H*_{n-1}(P), *r= rank H*_{n-1}(P).