The homology of cyclic branched covers of \(S^ 3\)
An important invariant of a knot in the 3-sphere is the homology \(H_1 (M_p)\) of its \(p\)-fold cyclic branched cover \(M_p\) (a closed orientable 3-manifold). The action of the cyclic covering group \(C_p = \langle g \rangle\) on \(M_p\) induces on \(H_1 (M_p)\) the structure of a \(\mathbb{Z} [C_p]\)-module and, for a prime \(p\), also the structure of a \(\mathbb{Z} [\zeta] \)-module where \(\zeta\) is a primitive \(p\)-th root of unity (noting that \(\mathbb{Z} [\zeta] = \mathbb{Z} [C_p]/ \Sigma\) because the norm element \(\Sigma = 1 + g + \cdots + g^{p - 1}\) annihilates \(H_1 (M_p))\). The main result of the present paper determines what abelian groups can arise as the first homology of a \(p\)-fold cyclic branched cover \(M_p\) of a knot, by giving necessary and sufficient conditions for a finitely generated \(\mathbb{Z} [\zeta]\)-module to be isomorphic to \(H_1 (M_p)\). The characterization is in terms of cokernels of Hermitian matrices over the Dedekind domain \(\mathbb{Z} [\zeta]\), and also in terms of ideal class invariants of \(\mathbb{Z} [\zeta]\)-modules (which are elements of the ideal class group of \(\mathbb{Z} [\zeta])\). Similar results are proved also for links; in this case there remains some gap between the given necessary and sufficient conditions. The paper is in large parts algebraic proving in particular a theorem about the possible modules occuring as cokernels of Hermitian matrices over Dedekind domains, under some hypotheses.
scientific article; zbMATH DE number 759280