Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

1. When constructing the Riemann surfaces, why does the space of germs construction correspond to the gluing construction?
2. Why is the connected component of the space of germs over $\mathbb{C}^*$ corresponding to the complex logarithm analytically isomorphic to $\mathbb{C}$?

I have read the page with this related question About the Riemann surface associated to an analytic germ which is basically asking a similar thing, so read it please if you find some of the terms I'm using unclear.

Whenever I've asked this questions to anybody, they have answered me with questions and "think about it" which is very annoying.

My view on this: Suppose we have a space of germs $G = \{[f]_z: z \in G\}$ over a domain $D$ ($\mathbb{C}^*$ in case of the complex logarithm function). Then we define a projection map $$p: G \rightarrow D$$ $$p:[f]_z \rightarrow z$$ $$p:[f]_A \rightarrow A$$ where $A$ is a domain in $D$. I'm able to show that $p$ is continuous and a homeomorphism. It seems clear that} $p$ is also a bijection.

Furthermore, we can also find a unique equivalence class of atlases on $G$ so that the projection $p$ is analytic.

$p$ is then analytic and a bijection (and a homeomorphism). Can we take $p$ as our analytic isomorphism then? Do we need to show $p^{-1}$ analytic (?) or are we done already? Or do we need something completely else altogether?

Can also define an evaluation map $\varepsilon: G \rightarrow \mathbb{C}$ with $$\varepsilon: [f]_z \rightarrow f(z)$$ which can be shown to be analytic as a map between Riemann Surfaces. Does this then answer the second question?

I'm suspicious of my answers above since they don't contain a word about "cut & paste" constructions (gluing) mentioned in the original problem. Anyone care to share their thoughts about this?

• Did you try reading Springer's book "Introduction to Riemann surfaces" mentioned in the 2nd answer in the link? Springer spends the whole chapter 3 addressing just this issue. How can you expect to get a more detailed answer here at MSE? – Moishe Kohan Mar 30 '14 at 5:24
• In the same vein as studiousus response, these exact questions are answered in Freitag's book Complex Analysis 2, chapter 1, section 2. – Alex Youcis Mar 30 '14 at 7:02
• @studiosus, I'm going to get that book in the library tomorrow. Is it wrong of me to ask a question on MSE, huh? – user138878 Mar 30 '14 at 12:23