Abstract
We calculate the first Betti number of an Abelian covering of a CW-complex X as the number of points with cyclotomic coordinates (of orders determined by the Galois group) which belong to a certain subvariety of a torus constructed from the fundamental group of X. This generalizes the classical formulas for the cyclic coverings due to Zariski and Fox. We also describe certain properties of these subvarieties of tori in the case when X is a complement to an algebraic curve in CP 2 which are analogs of Traldi–Torres relations from the link theory and the divisibility theorem for Alexander polynomials of plane algebraic curves [8].
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