This Problem Sheet is based on Lectures 22 and 23.

  • Problem 1 and 2 are about singular cubes.
  • Problem 3 is a nice statement about symplectic manifolds.
  • Problem 4 asks you to find a closed non-exact form on $ \mathbb{R}^n \setminus \{0\}$.
  • Problem 5 fills in a gap from today’s lecture.
  • Problem 6 investigates how diffeomorphisms affect integrals.
  • Problem 7 shows that on Lie groups we can also integrate functions.
  • Problem 8 is lots of fun. This introduces the degree of a smooth map between compact oriented connected smooth manifolds, and then uses this to prove the infamous Hairy Ball Theorem.

Comments and questions?