Thread: A question on Riemann surfaces and spaces of germs

1. A question on Riemann surfaces and spaces of germs

Hello everyone, I would really, -really- appreciate some detailed help on this question: I am taking a lecture course on Riemann Surfaces, and the lecturer has failed to explain how to approach a question like this at all unfortunately - plenty of theorems (monodromy, existence of lifts etc.) but nothing in the way of examples on how to approach a question like this.

Once I'm done with this problem I've got 4 or 5 more along very similar lines, and given that I'd like to try and get those done completely by myself following this, I would really appreciate as much detail as you can possibly give me. Obviously I will be using my own mathematical knowledge/intuition, but the more detail & help I can get from you for this one question, the more likely I am to grasp the concept for subsequent ones.

The question is as follows: "Show that the component of the space of germs over $\displaystyle \mathbb{C}^{*}$corresponding to the complex logarithm is analytically isomorphic to the Riemann surface constructed by gluing, and hence also analytically isomorphic to $\displaystyle \mathbb{C}.$"

Now I believe I can choose a 'base point' somewhere in the surface constructed by the gluing and then use analytic continuation to extend this to any given point - I am also fairly sure that whatever path we choose to extend it, we should get the same end result presumably, because we need the map to be well-defined - I suspect this follows from the Monodromy theorem.

However, I don't know where to go from here. Obviously we can project the Riemann surface down onto $\displaystyle \mathbb{C}^{*}$ by 'flattening it', but at the same time I believe the surface is simply connected (its complement is just {0} which is connected), and my thoughts are something along the lines of mapping from a point on the surface to the germ given by analytic continuation of the logarithm from our arbitrary fixed basepoint to the point on the surface - however, I'm uncertain as to how to formalize this argument, among other issues.

The question obviously refers to the fact that the complex logarithm is a multifunction, since we construct the Riemann surface by identifying the distinct branches of the logarithm and gluing them together. I believe each point in $\displaystyle \mathbb{C}^{*}$ should have a set of germs corresponding to $\displaystyle \mathbb{Z}$, one for each 'sheet' of the construction. So how would you actually construct the analytic isomorphism, and show it is analytic and isomorphic then? The way I picture the glued surface is actually -already- as $\displaystyle \mathbb{Z}$ copies of $\displaystyle \mathbb{C}^{*}$, or rather as $\displaystyle \mathbb{C}^{*} \times \mathbb{Z}$, but now it appears that is the structure of the space of germs - however, I find it hard to believe that my map is from (z,n) to the germ corresponding to $\displaystyle \log(|z|)+2n\pi i +i\theta$, where $\displaystyle 0 \leq \theta = arg(z) < 2\pi$ - that seems almost -too- obvious in a way, and I suspect I've made an incorrect assumption which has trivialized a non-trivial question.

For one thing, how can a map be analytically isomorphic onto the space of germs? Aren't germs (as far as we've got in the course) considered to be essentially functions at a point under the equivalence relation f~g at the point x iff they are identical on some open neighbourhood of x? So how can something be analytic onto such a space? I think part of the issue is that I am being expected (sadly!) to hand this work in before the lecturer is completely done covering the topic - so how would you go about a problem like this? As much as the mathematical content, actually knowing -what- to write ('can construct a function f between ___ and ___ s.t.' etc... - I don't want you to write it all out for me though, obviously!) is unclear to me too, so if you would be able to provide me with an example of how something like this should be properly attacked, that would be incredibly helpful. (The next question, for example, discusses the situation with the function $\displaystyle (z^3-z)^{(1/2)}$, and I would like to give that a go ASAP, so please respond soon if you can - apologies for the essay, it's just really key for my learning of the course that I grasp this concept .)

2. You may need to put more context here from your book. If this "germ" is really the equivalent class of local functions, what is the topology defined for this space? I'm wondering what a germ is in your context. And what is your C^{\star}? Is it C \ {0} ?

3. Originally Posted by xxp9
You may need to put more context here from your book. If this "germ" is really the equivalent class of local functions, what is the topology defined for this space? I'm wondering what a germ is in your context. And what is your C^{\star}? Is it C \ {0} ?
To me, a germ at the point z is an equivalence class of function elements (Function Element -- from Wolfram MathWorld), where (f,U)~(g,V) if the intersection of U and V is nonempty (contains, say, z), and f agrees with g on some neighbourhood of the point z contained in the intersection of U and V: so a germ is given at each point by locally identical function elements, call this [f]_z. Call [f]_D the union of all function elements at every point in the set D, i.e. the union of [f]_z for z in D.

The open sets are unions of sets of the form [f]_D, which I believe form a basis for the topology. This topology is Hausdorff, if that's relevant. And yes, C^{\star} is C \ {0}. Is that sufficient information now? Thanks!

4. Fix a point z, two locally constant functions f=c1 and g=c2 belong to two germs [f]_z and [g]_z. That is, two distinct points in the space of germs. Can you choose two open sets in this space, to separate these two points? Provided that you said it is Hausdorff.

5. Originally Posted by xxp9
Fix a point z, two locally constant functions f=c1 and g=c2 belong to two germs [f]_z and [g]_z. That is, two distinct points in the space of germs. Can you choose two open sets in this space, to separate these two points? Provided that you said it is Hausdorff.
Yes, I believe that any valid [f]_D, [g]_D (for the same D) should work, since these are disjoint - any element in both must be locally constant and equal to both constant functions around some z in D which is of course impossible. How does that help us though, sorry?

6. but in your definition, [f]_D is the set of all function elements for every z in D. so [f]_D=[g]_D in your term.

7. Originally Posted by xxp9
but in your definition, [f]_D is the set of all function elements for every z in D. so [f]_D=[g]_D in your term.
Sorry, I was unclear! When I said 'all function elements', I meant that for any specific function element (f,D), an open set is the union of sets of the form $\displaystyle [f]_D$, where $\displaystyle [f]_D$=$\displaystyle \{[f]_z: z \in D \}$, perhaps that's clearer? Thanks for persevering I think that if we take the projection map $\displaystyle \pi$ from the space of germs given by $\displaystyle \pi([f]_D)=D$, we will get an analytic map onto the Riemann Surface obtained by our gluing... Is that right? Seems very simple!

8. Still I'm not sure whether the topology is well defined or not. An example concerning me is a path c : [0,1] -> X where X is the space of germs. The path is defined by c(t) = [t]_z for a fixed point z, that is, t is mapped to a locally constant function. [t]_D is an open set containing [t]_z, but its pre-image {t} is not an open set of [0,1], which means this path is nowhere continous.
While this example doesn't really harm since there is no contradiction introduced, though it looks weird.
OK so let's continue on your idea. Now you have a projection $\displaystyle \pi$, but we need an analytic isomorphic, which is a bijection, how do you construct that?