Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 5 March 6, 2014

Problem 1: Show that for a differentiable manifold M and p ∈ M the map

E(M) ⟶ E_{p}(M), f ↦ f_{p}

which maps a function to its germ at p is surjective.

Problem 2:

(a) Show that for each n there exists a differentiable map f : R ⟶ R^{n} so that for each k ∈ N the set {f(t) | t ≥ k} contains all points for which all coordinates are rational.

(b) Find a function δ : R ⟶ R, δ > 0, and for each n a differentiable map f : R ⟶ R^{n}, so that for no embedding g : R ⟶ R^{n} one has |g − f| < δ

Problem 3: For a compact manifold M^{m} it is easy to prove an embedding theorem without regard to the dimension. One chooses a finite good atlas \{ϕ_{i} | i = 1, …, r\}, a bump function ψ for \overline{B_{1}(0)} with support in \overline{B_{2}(0)} and one sets ψ_{i} = ψ ∘ ϕ_{i} : M ⟶ R and k_{i} = ψ · ϕ_{i} : M ⟶ R^{m}.

Show that the map

M ⟶ ∏_{i=1}^{r} R^{m} × ∏_{i=1}^{r} R, p ↦ (k_{1}(p), … k_{r}(p), ψ_{1}(p), …, ψ_{r}(p))

is an embedding without using any further results from the lecture.

Problem 4: Let X, Y and Z be differentiable vector fields on a differentiable manifold M.

(a) Prove that [X, Y] defined by [X, Y]_{p}(f_{p}) = X_{p}(Y (f_{p})) − Y_{p}(X(f_{p})) for f_{p} ∈ E_{p}(M) is indeed a vector fieldon M.

(b) For f, g ∈ E(M) show that [fX, gY] = fg[X, Y] + fX(g)Y − gY (f)X.

(c) Show that [X, Y] = −[Y, X].

(d) Show that [X, [Y, Z]] = [[X, Y], Z] − [Y, [X, Z]].