The homology of cyclic and irregular dihedral coverings branched over homology spheres (Q1821387)

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.