some questions on Riemann surface

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There are several puzzling questions on Riemann surface for me: Q.1 Definition of Riemann surface can be given in at least two ways: Def.1) it is a complex one dimensional manifold; Def.2) for each $a\in \mathbb{C}$, consider collection of germs at $a$, of analytic functions, and give a topology on it. Are these really equivalent definitions? or Def.2 is more general than Def.1?

Q.2 When we say a group $G$ is an automorphism group of a compact Riemann surface, how is the action? (for ex. what is description of action of of PSL(2,7) on a genus 3 Riemann surface? In the book of Thomas Breuer, I couldn't see any description of action of a group on a Riemann surface; he has given computational methods to investigate groups.)

Q.3 The automorphisms of a compact Riemann surface can always be lifted to universal cover?

Q.4 If a group $G$ acts on a compact Riemann surface $X_g$, of genus $g$, then $X_g/G$ is also a compact Riemann surface of some genus $h$ and $g,h$ are related by Riemann-Hurwitz formula. Can anyone suggest some good reference for this relation? (here, I would like to see this Riemann Hurwitz relation topologically; many books describe it using algebraic geometry techniques).

(I went through many books on Riemann surface for these questions; but not understood many things)

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I can recommend for all those questions the book "Lectures on Riemann surfaces" by Otto Forster. One doesn't have to know any algebraic geometry to understand the book. – Someone Feb 10 2011 at 13:20
Regarding Q4: You can find a purely topological derivation of Riemann-Hurwitz in paragraph 21 of Prasolov & Sossinskys "Knots, links, braids and 3-manifolds". – bavajee Feb 10 2011 at 14:13
Your definition 2 does not seem to define anything at all, since you didn't specify additional conditions on the topology. Do you plan to choose some set-theoretic bijection between the points $a \in \mathbb{C}$ and the points on your Riemann surface? – S. Carnahan♦ Feb 10 2011 at 15:54
The first one is explained pretty well in Weyl's "The Idea of a Riemann Surface" (though the terminology is somewhat old-fashioned). – arsmath Feb 11 2011 at 12:08

2 Answers

Q1. There are two DIFFERENT notions of Riemann surface in the literature.

a) One-dimensional complex analytic manifold (coming from the book of Weyl).

b) Riemann surface "spread over the plane (or over the Riemann sphere)". Your second definition, the set of germs with an appropriate topology on it, formalizes this second notion.

Older books seem to understand Riemann surfaces in the sense of the second definition. Sometimes a) was called an "abstract Riemann surface" in these books.

For most mathematicians with modern training the "Riemann surface of log z" and the "Riemann surface of arccos z" are meaningless expressions because these are the same as the plane, in the sense of definition a).

The formal relation between a) and b) is the following. "A Riemann surface spread over the plane" is a pair (S,f), where S is an abstract Riemann surface and f is a holomorphic function from S to C. (If f is meromorphic, we have a Riemann surface spread over the sphere.)

Here is another way to say this. Let S be a Riemann surface in the sense a). It has a set of charts $\phi_j: U_j\to D_j$ from the elements of an open covering U to discs D in the plane. The correspoddence maps $\phi_k\circ\phi_j^{-1}$ on $D_j\cap D_k$ must be conformal.

Now let us require that these correspondence maps be IDENTITY maps of $D_j\cap D_k$. Then we obtain notion b). This is an additional structure on a Riemann surface in the sense a) which is sometimes called a flat structure.

If you look carefully (say, on the example of arccos) you will see that the two definitions of a Riemann surface in the sense b) that I gave are not exactly equivalent. More about this in my survey "Geometric theory of meromorphic functions", and in the preprint of Biswas and Perez Marco, Log Riemann Surfaces.

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Q1: Use the manifold definition, going back to Weyl. The other definition comes out of the theory of analytic continuation. (And is somewhat puzzling historically - I'm not quite sure how the Poincaré-Volterra theorem fits in, but these days you'd probably want to read this material in terms of sheaf theory, to which it was one of the inputs.)

Q2: G acts on the field of meromorphic functions, is one way to look at it. These are holomorphic mappings of the surface to itself, described by some algebraic mappings in fact.

Q3: I think so, by "abstract nonsense".

Q4: The quotient is to be treated carefully, since quotients of manifolds are not always manifolds. But in terms of the function field this can be seen as Galois theory, and X is a ramified (usually) covering of the quotient curve. The topological explanation of the Euler characteristic in the Riemann-Hurwitz formula is intuitively clear: just look at what happens under the k-th power map on the unit complex disc, in terms of a simple triangulation, to see how ramification affects coverings.

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For Q3: If H is the upper half-plane, S the Riemann surface, and pi:H->S, then composing pi with any automorphism gives a holomorphic map from H to S. Since this map takes a simply-connected surface to S (in particular the image of pi_1 of H is trivial), it must lift to a holomorphic map to H.  – Jonah Feb 11 2011 at 16:48
"lifting" the holomorphic map is easy; we apply a topological criteria; but is it easy to show that the lift is "holomorphic"? – Martin David Feb 12 2011 at 3:12
Sure. Say the automorphism of the surface you start with is called f, and the lifted automorphism is called F, and pick x in H. Pick a small enough neighborhood of F(x) so that the restriction of the (holomorphic) covering map is an analytic isomorphism. Now it is clear, since f compose pi is the same as pi compose F, that F can be written, in a suitably small neighborhood of x, as the composition of holomorphic maps. – Jonah Feb 12 2011 at 9:05
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