Middle East Technical University Fall 1995 Math 471 Elementary Topology and Geometry

MIDTERM EXAMINATION

Problem 1. Let K be the complex of a 3-symplex [v_{0}v_{1}v_{2}v_{3}].

(1) Determine ∂α if α = 2[v_{0}v_{2}v_{1}] − [v_{3}v_{2}v_{0}] ∈ C_{2}(K; ℤ);

(2) for β = [v_{0}v_{1}]−[v_{2}v_{1}]−[v_{2}v_{3}]+[v_{3}v_{0}] determine if β is a cycle in C_{1}(K; G) where (a) G = ℤ/3, (b) G = ℤ/2;

(3) is the above β a boundary ?

Problem 2. Describe an atlas consisting of 4 charts and which determine a differential structure on S^{1} × S^{1}. Write the coordinate change functions between one of these charts and the others.

Problem 3. Let f: ℂ → ℂ is defined by f(z) = z^{2} + z − 2. Determine if f can be extended to a smooth map of Riemannian spheres, ℂp^{1} → ℂp^{1}. Hint: Write the map f in the other chart in ℂp^{1}.

Problem 4. Consider the map f: ℝp^{2} → ℝp^{1}, [x:y:z] ↦ [x^{2}+y^{2}+z^{2} : xy+xz+yz]. Write f in some coordinate systems in ℝp^{2} and in ℝp^{1}.

Problem 5. Let point p ∈ ℝp^{2} has coordinates (2, 3) in the chart φ_{x}: U_{x} → ℝ^{2}, φ_{x}[x : y : z] = (\frac{y}{x}, \frac{z}{x}), where U_{x} = \{[x : y : z] | x ≠ 0\}.

(1) Determine coordinates of p in the other two charts, φ_{y} and φ_{z}.

(2) For the vector v ∈ T_{p}(ℝp^{2}) which has coordinates (1, −1) in the chart φ_{x} determine coordinates in the chart φ_{y}.