Middle East Technical University Fall 1998 Math 541 Differentiable Topology

PROBLEM LIST 2

Critical and regular points, transversality and Sard’s theorem

Problem 1. Show that if M ⊂ ℝ^{p} is a differentiable submanifold, n ⩾ 1, then there exists a hyperplane, H ⊂ ℝ^{p}, M ∩ H ≠ ∅, which cuts M transversally.

Problem 2. Let M be a compact manifold, f: M → ℝ^{n+1} a differentiable function and 0 ∉ f(M). Show that there exists a line through the origin of ℝ^{n+1}, which only meets finitely many points of f(M).

Problem 3. Let f: S^{2} → ℝ^{2} be a smooth map. Prove that for an open, dense, subset of points, x ∈ f(S^{2}), the number of points in f^{−1}(x) is ⩾ 2.

Problem 4. Given a smooth mapping f: X → Y, between manifolds, prove that f is constant on the connected components of the critical point set, C ⊂ X.

Degrees of maps

Problem 5. Prove that the degree of a composition g ∘ f is equal to the product (deg g)(deg f).

Problem 6. Let W be a compact connected oriented (k + 1)-manifold with boundary, and let f: W . → S^{k} be a smooth map. Prove that f extends to a map F : W → S^{k} if and only if deg f = 0.

Problem 7. Show that any mapping f: S^{n} → S^{n} of degree deg f ≠ (−1)^{n+1} must

have a fixed point. (Hint. Prove the following Lemma: if maps f, g: X → S^{n} satisfy the condition ||f(x) − g(x)|| < 2, for any x ∈ X, then f is homotopic to g.)

Problem 8. Given a pair of disjoint manifolds M, N ⊂ ℝ^{k+1}, the linking map

λ: M × N → S^{k} is defined as
λ(x, y) = \frac{x − y}{||x − y||}

If M, N are closed, oriented, with the total dimension m + n = k, the the degree of λ is called the linking number, lk(M, N). Prove that

(1) lk(N, M) = lk(M, N),

(2) If M bounds an oriented manifold X disjoint from N, then lk(M, N) = 0.