Tanabe, Masaharu A homological characterization of surface coverings. (English) Zbl 1160.14017 Proc. Japan Acad., Ser. A 84, No. 9, 170-173 (2008). 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. Reviewer: Aaron Wootton (Portland) MSC: 14H30 Coverings of curves, fundamental group 30F30 Differentials on Riemann surfaces Keywords:surface coverings; homology groups; Riemann surfaces Citations:Zbl 0770.14018; Zbl 0158.07803 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] R. D. M. Accola, Automorphisms of Riemann surfaces, J. Analyse Math. 18 (1967), 1-5. · Zbl 0158.07803 · doi:10.1007/BF02798030 [2] L. V. Ahlfors and L. Sario, Riemann surfaces , Princeton Univ. Press, Princeton, N.J., 1960. · Zbl 0196.33801 [3] H. Hopf, Beiträge zur Klassifisierung der Flächenabbildungen, J. Reine Angew. Math. 165 (1931), 225-236. · Zbl 0002.05602 · doi:10.1515/crll.1931.165.225 [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 · doi:10.1090/conm/136/1188203 [5] H. H. Martens, A footnote to the Poincaré complete reducibility theorem, Publ. Mat. 36 (1992), no. 1, 111-129. · Zbl 0780.30031 · doi:10.5565/PUBLMAT_36192_09 [6] H. Poincaré, Sur la réduction des intégrales abéliennes, Bull. Soc. Math. France 12 (1884), 124-143. · JFM 16.0426.03 [7] H. Poincaré, Sur les fonctions abéliennes, Amer. J. Math. 8 (1886), 239-342. [8] G. Springer, Introduction to Riemann surfaces , Addison-Wesley, Inc., Reading, Mass., 1957. · Zbl 0078.06602 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.