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Let $f$ be a non-constant entire function. I'd like to view $f$ as the projection mapping in a fiber bundle, where the base space is $X =f(\mathbb{C})$ (which according to Picard is either $\mathbb{C}$ or $\mathbb{C} \setminus \{z_0\}$), and the bundle space is the collection $\bigcup_{x\in X}f^{-1}(x)$. In words, the fiber over each $f(z)=x\in X$ is the (discrete) set of points which $f$ maps to $x$. Is anyone familiar with a reference treating this aspect ?

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A pedantic remark is that it won't always be a fiber bundle (which in this case is more commonly called a covering space as the fibers will be discrete) over the image. For instance, if $f(z) = z^2$, the fiber over the origin doesn't look like the other fibers.

Anyway, you are describing the Riemann surface attached to the problem of inverting $f$. For example, if $f(z) = z^2$ you are describing the "Riemann surface of the square root function."

http://en.wikipedia.org/wiki/Riemann_surface

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Thanks. I understand the standard treatment of the inverse function is done using the Riemann surface. I was wondering whether the fiber bundle approach (or covering space as you mention) can add something, and whether there's a reference which investigates this approach. –  Teddy Aug 13 '12 at 15:04
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