JournalscmhVol. 78 , No. 4DOI 10.1007/s00014-003-0781-x

Analogies between group actions on 3-manifolds and number fields

  • Adam S. Sikora

    University at Buffalo SUNY, USA
Analogies between group actions on 3-manifolds and number fields cover

Abstract

Let a cyclic group GG act either on a number field L\mathbb L or on a 33-manifold MM. Let sLs_{\mathbb L} be the number of ramified primes in the extension LGL\mathbb L^G\subset \mathbb L and sMs_M be the number of components of the branching set of the branched covering MM/GM\to M/G. In this paper, we prove several formulas relating sLs_{\mathbb L} and sMs_M to the induced GG-action on Cl(L)Cl(\mathbb L) and H1(M),H_1(M), respectively. We observe that the formulas for 33-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between 33-manifolds and number fields postulated in arithmetic topology.