Middle East Technical University Spring 1994 Math 471

FINAL EXAM

JUNE, 15, 1994

Problem 1. (5 pts) Show that if X is simply connected (i.e., if any closed curve in X bounds a disc) then ∮_{γ} ω = 0 for all closed curves γ in X and closed forms ω on X.

Problem 2. (5 pts) Suppose that X = ∂W, W is compact and f: X → Y is a smooth map. Let ω be a closed k-form on Y, where k = dim X. Prove that if f extends to all W, then ∫_{X} f^{∗}ω = 0.

Problem 3. (5 pts) Consider 1-form ω = \frac{du}{1+u^{2}} in ℝ as 1-form in ℝp^{1} given in the chart h: \{(x : y) ∈ ℝp^{1}, y ≠ 0\} → ℝ , h(x : y) = \frac{x}{y}. Prove that ω can be extended smoothly to the whole ℝp^{1}. Will it be true for ω = du ?

Problem 4. (5 pts) Prove that a 1-form ω on a circle is a differential of a function if and only if ∫_{S^{1}} ω = 0.

Hint: use the 1-form f^{∗}ω on ℝ induced by the map f: ℝ → S^{1}, f(t) = e^{2πit}.