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On the Iwasawa \(\mu\)-invariants of branched \(\mathbf Z_p\)-covers. (English) Zbl 1358.57009

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 Y. Furuta [Nagoya Math. J. 29, 281–285 (1967; Zbl 0166.05901)] for number fields. In addition the author formulate analogues of 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.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
11R23 Iwasawa theory
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References:

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