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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    $\begingroup$ -1. Please add more details. (It isn't hard to write a homology basis of such a double cover using a cell decomposition, though.) $\endgroup$ – S. Carnahan Apr 2 '10 at 21:51
  • $\begingroup$ I'm refraining from downvoting, but this is a good example of a question which could be better written - providing more motivation or context $\endgroup$ – Yemon Choi Apr 2 '10 at 22:33
  • $\begingroup$ 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. $\endgroup$ – S. Carnahan Apr 2 '10 at 22:39
  • $\begingroup$ 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. $\endgroup$ – Dror Speiser Apr 2 '10 at 23:39

Here is a vague suggestion as to why there might not be a "canonical" choice of basis. I will presume that you already have in mind a choice of basis for the homology of your elliptic curve. Write Gamma_{1,2} for the group of isotopy classes of oriented diffeomorphisms of the twice-punctured elliptic curve (i.e. the mapping class group) and Gamma_1 = SL_2(Z) for the mapping class group of the unpunctured elliptic curve. Then the natural surjection Gamma_{1,2} -> SL_2(Z) has a kernel G, which can be thought of as the braid group on two strands on the torus (I'm sure Tom can say more than I can, offhand, about what this group is.)

Now G acts on H_1(X,Z), where X is your spectral curve, and I think for your basis to be "canonical" it would want to be fixed by this. But I don't immediately see why this action would be trivial. (Of course, by construction it acts trivially on the natural quotient H_1(E,Z).)


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