(1) Let N be the north pole of S2 and S be the south pole of S2

1. Homework 1

Let S²= {(a1, a2, a3) ∈ R³: a²₁+ a²₂+ a²₃= 1} be the closed unit sphere in R³. Equip S² with the subspace topology induced from R³, i.e. a subset V of S²is open in S²if and only if there exists an open subset U of R³so that V = U ∩ S².

(1) Let N = (0, 0, 1) be the north pole of S² and S = (0, 0, 1) be the south pole of S². Define two sets UN and US by UN = S²\ {S} and US = S²\ {N }. Prove that UN and US are open subsets of S² so that UN∪ US = S², i.e. {UN, US} forms an open cover for S².

(2) Define two functions ψ_N : U_N → C and ψS : U_S → C by ψS(a1, a2, a3) = a₁

1 − a3

+ i a₂ 1 − a3

, ψN(a1, a2, a3) = a₁ 1 + a3

− i a₂ 1 + a3

.

Prove that ψN and ψSare both homeomorphisms on S²such that ψN(N ) = 0 and ψS(S) = 0.

Verify that {(US, ψS), (UN, ψN)} is a C^∞-atlas for S².

(3) Let D be any open subset of C. A smooth function f : D → C is holomorphic on D if

∂f

∂z = 1 2

 ∂f

∂x + i∂f

∂y



= 0 on D.

Find ψ_N(U_N ∩ U_S) and ϕ_S(U_N ∩ U_S). (They are open.) Check that the transition function ψS◦ ψ⁻¹_N : ψN(UN∩ US) → ψS(UN ∩ US)

is holomorphic i.e. check that A = {(U_S, ψ_S), (U_N, ψ_N)} is a complex atlas on S². What happen if we replace −i by +i in ψN? Is {(US, ψS), (UN, ψN)} a complex atlas on S²? (4) Let f : S² → C be a holomorphic function. Prove that f is a constant function. (See

Theorem 1.36 and Theorem 1.37) This implies that O(S²) ∼= C.

(5) Let f : US → C be the function

f (a1, a2, a3) = a²₁− a²₂+ 2a1a2i 1 − 2a3+ a²₃ . Prove or disprove that f is a holomorphic function on US.

(6) Let F : S²→ S² be the function sending (a1, a2, a3) to (−a1, −a2, −a3). The function F is called the antipodal map on S². Is F a holomorphic map on S²? (See definition 3.1 on page 32) Is F an isomorphism of Riemann surface (an automorphism of S²)? (See definition 3.6.

on page 46).

(7) Find a meromorphic function on S².

1

2

2. HW2

Let X be a nonempty set. An equivalence relation on X is a subset R of X × X with the following properties:

(1) (x, x) ∈ R for any x ∈ X;

(2) if (x, y) ∈ R, (y, x) ∈ R;

(3) if (x, y) ∈ R and (y, z) ∈ R, (x, z) ∈ R.

The equivalent class of x ∈ X is the set [x] = {y ∈ X : (x, y) ∈ R}. The set of all equivalent classes of elements of X is denoted by X/R and called the quotient set. The function

π : X → X/R, x 7→ [x]

is called the quotient map.

(A review of quotient topology) Let (X, τX) be a topological space and R is an equivalence relation on X. Let

τ_X/R= {U ⊆ X/R : π⁻¹(U ) ∈ τX}.

(1) Show that τ_X/R forms a topology on X/R.

(2) Show that the quotient map π : (X, τX) → (X/R, τ_X/R) is continuous.

(3) Suppose τ_X/R⁰ is another topology on X/R containing τ_X/Rso that π : (X, τX) → (X/R, τ_X/R⁰ ) is continuous. Show that τ_X/R⁰ = τ_X/R. In other words, τ_X/R is the largest topology so that π is continuous.

Let C be the field of complex numbers equipped with the standard Euclidean topology. Let {ω1, ω2} be two complex numbers so that {ω1, ω2} is linearly independent over R. Let

L = {m₁ω₁+ m₂ω₂∈ C : m1, m₂∈ Z.}

Then L is an abelian subgroup of (C, +). Two complex numbers z¹ and z2 are said to be equivalent if z₁− z2∈ L. In this case, we write

z₁≡ z2 mod L.

The quotient set C/L has a structure of abelian group defined by [z1] + [z2] = [z1+ z2].

We equip C/L the quotient topology.

(1) Let V be any open subset of C. Show that π⁻¹(π(V )) = [

ω∈L

(ω + V )

and that π(V ) is an open subset of C/L. (Hence π is an open mapping).

(2) Show that there exists  > 0 such that |ω| > 2 for all ω ∈ L \ {0}. For each z₀ ∈ C, show that

π : B(z⁰, ) → π(B(z⁰, )) is a homeomorphism. Here B(z0, ) = {z ∈ C : |z − z0| < }.

(3) For each z ∈ C, let U^z = π(B(z⁰, )) and ϕz : Uz → B(z, ) be the function ϕz = π|⁻¹

B(z0,). Suppose that Uz∩ Uw is nonempty, let T : φz(Uz∩ Uw) → φw(Uz∩ Uw) be the function T = φw◦ φ⁻¹_z .

(a) Show that π(T (z)) = π(z) for all z ∈ φz(Uz∩ Uw).

(b) By (3a), T (z) − z ∈ L for any z ∈ φz(Uz∩ Uw). Define a function g : φz(Uz∩ Uw) → L by g(z) = T (z) − z. Show that g is a locally constant function.

(c) By (3b), for any z0 ∈ φz(Uz∩ Uw), we can find an open neighborhood U of z0 and ω ∈ L so that g(z) = ω for all z ∈ U. In this case, T (z) = z + ω for all z ∈ U. Conclude that T is a holomorphic function on φ_z(U_z∩ Uw).

(4) Conclude that {(U_z, ϕ_z)} is a complex atlas on C/L so that C/L is a complex manifold of dimension one.