Find the fundamental group of a Riemann Surface

[phasic]

Homework Statement

χ is the Riemann Surface defined by P(w, z) = 0, where P is a complex polynomial of two variables of degree 2 in w and of degree 4 in z, with no mixed products. Find the fundamental group of χ.

Homework Equations

A variation of the Riemann-Hurwitz Formula states that if χ is a compact RS defined by P(z,w) = 0, max degree(w) = p > 0, and genus, g ≥ 0, then while b is the number of branching points of the surface, b = 2*(g + p - 1)

The Attempt at a Solution

Using the formula, p = 2, b = 4, and 4 = 2*(g + 2 - 1). Thus, g = 1.

This implies the surface is topologically equivalent to the torus, with genus 1, which has the fundamental group ≈ ∏(S1)x∏(S1) ≈ ZxZ.

Am I right on this one? I'm not very good at visualizing these complex functions, but this formula sure makes it easy to know at least what the surface is homeomorphic to. Any tips on better understanding the surface would be nice, including software to model it.

 

[mighty2000]

Yes, your solution is correct. The fundamental group of the Riemann surface defined by P(w,z) = 0 is ZxZ, which is the same as the fundamental group of a torus.

One way to better understand the surface is to plot it using software such as Mathematica or Wolfram Alpha. This will give you a visual representation of the surface and its properties. You can also try to find a parametrization of the surface, which will help you understand how the surface is constructed and how it behaves.

Another useful tool is to use differential geometry concepts to study the surface, such as finding its curvature or geodesics. This will give you a deeper understanding of the surface and its properties.

Overall, the more you study and visualize Riemann surfaces, the better you will become at understanding and working with them.