On the homology of metacyclic coverings
We study the first homology group of irregular metacyclic coverings. Let G be a metacyclic group, i.e., an extension of \({\mathbb{Z}}/n{\mathbb{Z}}\) by \({\mathbb{Z}}/m{\mathbb{Z}}\), for some n,m. Assume that \(p: M\to \Sigma\) is an n- fold branched covering, where M and \(\Sigma\) are P.L. manifolds, with branch index m in the singular locus and monodromy group G. If \(\Sigma\) is simply connected our main result gives a congruence relation for dim \(H_ 1(M,F)\), F a field of characteristic \(\neq n\), in terms of n, m and a third number that determines G. If F is \({\mathbb{Q}}\) or a finite prime field whose characteristic is a generator of \(({\mathbb{Z}}/n{\mathbb{Z}})^*\) we find: \(\dim H_ 1(M,F)\equiv 0 mod((n-1)/m).\)
first homology group of irregular metacyclic coverings
scientific article; zbMATH DE number 3925803