A homological characterization of surface coverings

Abstract

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.