Homework 1 Complex Manifolds Math 241, Spring 1996, Berkeley CA C. McMullen Due Tuesday, 6 February 96

1. Let f : U → ℂ be an analytic function defined on disk U ⊂ ℂ, and let γ ⊂ U be a smooth simple closed curve. Show that

g(w) = \frac{1}{2πi} ∫_{γ} \frac{zf′(z) dz}{f(z) − w}

is an analytic function on ℂ − f(γ). What function is it (e.g. when f is injective)?

2. Prove that a proper local homeomorphism between locally compact Hausdorff spaces is a covering map.

3. Let X be a pair of pants, i.e. a compact surface with boundary diffeomorphic to S^{2} with three disjoint disks removed.

(a) What is π_{1}(X)?

(b) Draw a picture of the universal cover of X tiled by a fundamental domain.

(c) Draw a similar picture of the covering space corresponding to π_{1}(B) ⊂ π_{1}(X), where B is one of the boundary components.

4. Let X be a compact Riemann surface, and E ⊂ X a finite set. Let f : (X − E) → ℂ̂ be a nonconstant holomorphic function. Prove the image of f is dense.

5. How many roots of the equation z^{4} − 6z + 3 = 0 have modulus between 1 and 2?

6. Let f(z, w) = sin(z^{2}) − w. Find the Weierstass polynomial p(z, w) = z^{d}+a_{1}(w)z^{d−1}+…+a_{d} such that f(z, w) = p(z, w)h(z, w) near (z, w) = (0, 0), with h(0, 0) ≠ 0.

7. Find a value of τ such that ℂ/(ℤ⊕τℤ) and ℂ^{∗}/(z ↦ 2z) are isomorphic Riemann surfaces.

8. Find a finite set E ⊂ ℂ̂ such that ℂ̂ − E is conformally equivalent to a cube with its vertices removed.

9. (a) Find a power series f(z, w) = ∑ a_{ij}z^{i}w^{j} in ℂ^{2} whose region of absolute convergence is the bidisk Δ^{2} = {z, w : |z| < 1, |w| < 1}.

(b) Show there is a power series f(z, w) = ∑ a_{ij}z^{i}w^{j} in ℂ^{2} whose region of absolute convergence is the unit ball

B = {(z, w) : |z|^{2} + |w|^{2} < 1}.

(Hint: B is the intersection of all the unitary rotations of the bidisk.)