Commentarii Mathematici Helvetici

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Volume 78, Issue 4, 2003, pp. 832–844
DOI: 10.1007/s00014-003-0781-x

Published online: 2003-12-31

Analogies between group actions on 3-manifolds and number fields

Adam S. Sikora[1]

(1) University at Buffalo SUNY, USA

Let a cyclic group $G$ act either on a number field $\mathbb L$ or on a $3$-manifold $M$. Let $s_{\mathbb L}$ be the number of ramified primes in the extension $\mathbb L^G\subset \mathbb L$ and $s_M$ be the number of components of the branching set of the branched covering $M\to M/G$. In this paper, we prove several formulas relating $s_{\mathbb L}$ and $s_M$ to the induced $G$-action on $Cl(\mathbb L)$ and $H_1(M),$ respectively. We observe that the formulas for $3$-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between $3$-manifolds and number fields postulated in arithmetic topology.

Keywords: Arithmetic topology, cyclic group action, knot, prime

Sikora Adam: Analogies between group actions on 3-manifolds and number fields. Comment. Math. Helv. 78 (2003), 832-844. doi: 10.1007/s00014-003-0781-x