Annales Academię Scientiarum Fennicę
Mathematica
Volumen 34, 2009, 391-399

HURWITZ' THEOREM AND A GENERALIZATION FOR HOLOMORPHIC MAPS OF CLOSED RIEMANN SURFACES

Masaharu Tanabe

Tokyo Institute of Technology, Department of Mathematics
Ohokayama, Meguro, Tokyo, 152-8551, Japan; tanabe 'at' math.titech.ac.jp

Abstract. Let X be a closed Riemann surface of genus greater than one. Hurwitz showed that an automorphism of X is completely determined by the induced automorphism on H₁(X,Z). We study this theorem in the context of H¹(X,Z) and we prove the following as a generalization. Let \widetilde X,X₁,X₂ be closed Riemann surfaces of genera greater than one and let f_i : \widetilde X \to X_i (i = 1,2) be non-constant holomorphic maps. Assume that there exist a_i,b_i \in H¹(X_i,Z) (i = 1,2) so that \int\int_{X_i}a_i \land b_i = 1 (i = 1,2) and that f₁^*a₁ = f₂^*a₂ and f₁^*b₁ = f₂^*b₂ in H¹(\widetilde X,Z). Then there exists a conformal map h : X₁ \to X₂ which satisfies f₂ = h o f₁.

2000 Mathematics Subject Classification: Primary 30F30; Secondary 14F40.

Key words: Riemann surfaces, homology groups, de Rham cohomology.

Reference to this article: M. Tanabe: Hurwitz' theorem and a genararization for holomorphic maps of closed Riemann surfaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 391-399.

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