On the homology of metacyclic coverings
type: Item
instance of: scholarly article
title: On the homology of metacyclic coverings
zbMATH Open document ID: 0578.57001
DOI: 10.1007/BF01458590
author: Antonio F. Costas
author: Jesús M. Ruiz
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164140
review text: 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: \(\dim H_ 1(M,F)\equiv 0 mod((n-1)/m).\)
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 3925803
zbMATH Keywords: first homology group of irregular metacyclic coverings
zbMATH Keywords: branched covering
zbMATH Keywords: PL-manifold
MaRDI profile type: Publication
cites work: Q3236675
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description: scientific article; zbMATH DE number 3925803