Middle East Technical University Differential Topology Math 541 Fall 2004

Midterm II

(take-home)

Problem 1. Suppose that for a smooth mapping f: X → Y, d_{x}f = 0 at all points x ∈ X. Prove that f is constant on the connected components of X.

Problem 2. Prove that any embedded circle in ℝ^{n}, n ≥ 5, bounds an embedded disc.

Problem 3. Assume that f: X → Y is a submersion. Prove that any vector field, W, on Y can be lifted to X, that is to say, there exists a vector field V on X, such that d_{x}(V (x)) = W (f(x)), ∀x ∈ X.

Problem 4. Assume that s: X → E is a continuous section of a smooth Euclidian vector bundle p: E → X. Prove that for any ϵ > 0 there exists a smooth ϵ-approximation, s′: X → E, of s.