Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 6 March 13, 2014

Problem 1: (a) Show that for each natural number n ≥ 0 there is a flow on S^{1} with exactly n fixed points.

(b) Show that on the torus S^{1} × S^{1} there exists a vector field for which no orbit of the associated flow is a submanifold of S^{1} × S^{1}. Hint: S^{1} × S^{1} = (R × R)/(Z × Z). Consider a specific constant vector field on R^{2}.

Problem 2: Show that every submanifold diffeomorphic to S^{1} inside a differentiable manifold M arises as the orbit of a global flow on M. Hint: Partitions of unity.

Problem 3: Show that a bounded vector field defined on R^{n} is globally integrable.

Problem 4: Let E and F be vector bundles over X. Prove that the kernel and image of a bundle isomorphism f : E ⟶ F of constant rank are subbundles of E respectively F. Hint: One has to show that for each point p ∈ X there exists bundle charts (ϕ, U) for E and (ψ, U) for F such that for every x ∈ U the map ψ ∘ f ∘ ϕ^{−1} : U × R^{m} ⟶ U × R^{n} is given by (u, (v_{1}, …, v_{m})) ↦ (u, (v_{1}, …, v_{k}, 0, …, 0)). Use a method similar as was done for the local form of differential maps.