On the Iwasawa \(\mu\)-invariants of branched \(\mathbf Z_p\)-covers (Q315582)

The author establishes a relative genus theory for a branched cover of rational homology 3-spheres. The author defines the relative genus cover for a finite branched Galois cover of a 3-manifold \(M\) (oriented, connected and closed) as the maximal unbranched cover obtained as a composite of the original Galois cover and a branched abelian cover of \(M\). The main result is an explicit formula for the relative genus number similar to the one obtained by \textit{Y. Furuta} [Nagoya Math. J. 29, 281--285 (1967; Zbl 0166.05901)] for number fields. In addition the author formulate analogues of \textit{K. Iwasawa}'s theorems [in: Number Theory, algebr. Geom., commut. Algebra, in Honor of Yasuo Akizuki, 1--11 (1973; Zbl 0281.12005)] on \(\mu\)-invariants for branched \(\mathbb{Z}_p\)-covers of rational homology 3-spheres, by using relative genus theory.