The homology of cyclic and irregular dihedral coverings branched over homology spheres
type: Item
instance of: scholarly article
title: The homology of cyclic and irregular dihedral coverings branched over homology spheres
zbMATH Open document ID: 0616.57001
DOI: 10.1007/BF01456338
author: Valerio Chumillas
author: José María Montesinos-Amilibia
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164376
review text: We study the homology groups of the cyclic and of the irregular dihedral coverings branched over homology n-spheres. For each odd prime p, we describe a class \({\mathcal D}_ p\) of finitely generated abelian groups such that \(H_ i(M^ n, {\mathbb{Z}})\), \(i\neq 0,n\), belongs to \({\mathcal D}_ p\), for every p-fold irregular dihedral covering \(M^ n\) branched over a homology n-sphere \(N^ n\) with branch indices \(\leq 2\). We prove that an element of \({\mathcal D}_ p\) is isomorphic to such a \(H_ 1(M^ 3, {\mathbb{Z}})\) with \(N^ 3=S^ 3\), when the ring of integers of the real cyclotomic field \({\mathbb{Q}}[\xi +\xi ^{-1}]\), \(\xi =e^{2\pi i/p}\), is a principal ideal domain. This happens for all primes less than 68 (and if the Riemann hypothesis is true, for all primes less than 163). We obtain an analogous class \({\mathcal C}_ p\) to which \(H_ i(M^ n, {\mathbb{Z}})\), \(i\neq 0,n\), belongs if \(M^ n\) is a p-fold cyclic covering branched over a homology n-sphere, p prime.
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 3998778
zbMATH Keywords: branched over homology n-spheres
zbMATH Keywords: irregular dihedral branched covers
zbMATH Keywords: cyclic branched coverings
MaRDI profile type: Publication
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OpenAlex ID: W1973617801
description: scientific article; zbMATH DE number 3998778