# Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?

Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$, is there a standard choice of generators for $H_1(X, \mathbb{Z})$ in terms of the information about the branched points and the given basis?

The special case I have in mind is the spectral curve of some integrable system, and it is a double cover of a fixed elliptic curve with two branch points.

Are there good references for this?

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-1. Please add more details. (It isn't hard to write a homology basis of such a double cover using a cell decomposition, though.) – S. Carnahan Apr 2 '10 at 21:51
I'm refraining from downvoting, but this is a good example of a question which could be better written - providing more motivation or context – Yemon Choi Apr 2 '10 at 22:33
I might as well add a physical picture: Take a big sphere, and add two small handles near the equator in a way that preserves the symmetry under 180 degree rotation around the polar axis. – S. Carnahan Apr 2 '10 at 22:39
It's a reference question - how much clearer can it be? If you know of a book or online lecture notes on the topic, just post them. – Dror Speiser Apr 2 '10 at 23:39