• 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 $\mathbb{R}^m \setminus \{0\}$.
• Problem 5 is used in Lecture 27.
• 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.