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Find the fundamental group of a Riemann Surface

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant 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)


    3. 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.
     
  2. jcsd
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