Mathematics 730 Fundamental Concepts of Topology Homework Due Wednesday, October 21, 1998 Smooth Manifolds and Maps

For this problem, think of the 2-sphere S^{2} as a the one-point compactification ℂ∪{∞} of the complex plane.

1. Show that S^{2} has a structure of smooth 2-manifold, determined by the charts ϕ_{0} = id : ℂ → ℂ ≅ ℝ^{2} and ϕ_{∞} = (ℂ \ \{0\}) ∪ ∞ → ℂ ≅ ℝ^{2} given by ϕ_{∞}(z) = \frac{1}{z}. (Here we are using complex arithmetic and, by definition, \frac{1}{∞} = 0.)

2. Let f be a meromorphic function defined on the complex plane, in the sense of complex analysis. (That means that f is an analytic complex-valued function on ℂ \ D, where D is a discrete set of poles.) Show that if we define f to have the value ∞ at the set of poles D, then f becomes a smooth map ℂ ≅ ℝ^{2} → S^{2} (with respect to the smooth structure defined in (1)). Compute (in terms of the complex derivative f′) the differential of f as a map ℝ^{2} → ℝ^{2}, both at points where f is analytic and at the poles. What are the critical points and critical values?

3. The Weierstrass function ℘ is defined in complex analysis to be a doubly periodic meromorphic function with a single double pole in each fundamental domain. For present purposes, let the period lattice Λ be ℤ + iℤ (which is identified with ℤ^{2} when we identify ℂ with ℝ^{2} the usual way). Then, by definition,

℘(z) = \frac{1}{z^{2}} + ∑_{ω∈Λ\\{0\}} (\frac{1}{(z − ω)^{2}} − \frac{1}{ω^{2}} ) .

Show using (2) that ℘ induces a smooth map ℝ^{2}/ℤ^{2} ≅ T^{2} → S^{2}. Describe, as precisely as possible, the critical points and critical values of this map, and the nature of the inverse images ℘^{−1}(w) both when w is a regular value and when w is a critical value of the map.