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A homological characterization of surface coverings. (English) Zbl 1160.14017

Suppose that \(X\) and \(Y\) are compact Riemann surfaces of genera \(g\) and \(\gamma\) respectively and \(f: X\rightarrow Y\) is a holomorphic map. Then \(f\) induces a homomorphism \(f_{*} : H_{1}(X,{\mathbb Z}) \rightarrow H_{1}(Y,{\mathbb Z})\) on the first homology groups. Conversely however, given an abstract homomorphism \(\phi : H_{1}(X,{\mathbb Z}) \rightarrow H_{1}(Y,{\mathbb Z})\), it is not always the induced map of some holomorphic map \(f: X\rightarrow Y\). Therefore, it is natural to ask given an abstract homomorphism \(\phi : H_{1}(X,{\mathbb Z}) \rightarrow H_{1}(Y,{\mathbb Z})\) as a matrix, what conditions guarantee that \(\phi\) is the induced map of a holomorphic map \(f: X\rightarrow Y\). In the paper under review, the author considers this problem in the special case that the degree of \(f: X\rightarrow Y\) is of prime order.
The authors’ approach to the problem is to first start with a general form for the induced map \(f_{*}\) of a prime degree map \(f: X\rightarrow Y\) as determined in H. H. Martens [Contemp. Math. 136, 287–296 (1992; Zbl 0770.14018)]. The author then imposes additional restrictions on this general form using algebraic and analytical results, with the main result used coming from R. D. M. Accola [J. Anal. Math. 18, 1–5 (1967; Zbl 0158.07803)]. Using these techniques, the author shows that there are only two possible normal forms which guarantee \(\phi\) is an induced map when \(\gamma >1\), and just one for \(\gamma =1\). Moreover, for \(\gamma >1\), one of these forms occurs if and only if \(f: X\rightarrow Y\) is a normal unramified cover. The author is also able to give the explicit normal forms for all cases.

MSC:

14H30 Coverings of curves, fundamental group
30F30 Differentials on Riemann surfaces
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References:

[1] R. D. M. Accola, Automorphisms of Riemann surfaces, J. Analyse Math. 18 (1967), 1-5. · Zbl 0158.07803
[2] L. V. Ahlfors and L. Sario, Riemann surfaces , Princeton Univ. Press, Princeton, N.J., 1960. · Zbl 0196.33801
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[4] H. H. Martens, On the reduction of Abelian integrals and a problem of H. Hopf, in Curves, Jacobians, and abelian varieties ( Amherst, MA, 1990 ), 287-296, Contemp. Math., 136, Amer. Math. Soc., Providence, RI, 1992. · Zbl 0770.14018
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