Let $f:X\to Y$ be a covering of closed oriented surfaces. Then $f$ induces a homomorphism $f_{*} :H_{1} (X,\mathbf{Z})\to H_{1} (Y,\mathbf{Z})$ of the first homology groups. We consider the converse and characterize—in terms of matrices—the abstractly given homomorphisms of the first homology groups which can be induced by coverings of prime degree. We also classify the induced homomorphisms in these cases.
References
R. D. M. Accola, Automorphisms of Riemann surfaces, J. Analyse Math. 18 (1967), 1–5. MR213538 10.1007/BF02798030R. D. M. Accola, Automorphisms of Riemann surfaces, J. Analyse Math. 18 (1967), 1–5. MR213538 10.1007/BF02798030
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. MR1188203 10.1090/conm/136/1188203H. 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. MR1188203 10.1090/conm/136/1188203
H. H. Martens, A footnote to the Poincaré complete reducibility theorem, Publ. Mat. 36 (1992), no. 1, 111–129. MR1179606 10.5565/PUBLMAT_36192_09H. H. Martens, A footnote to the Poincaré complete reducibility theorem, Publ. Mat. 36 (1992), no. 1, 111–129. MR1179606 10.5565/PUBLMAT_36192_09
