This Problem Sheet is based on Lectures 12 and 13.

• Problem 1 shows that the (group-theoretic) kernel of a Lie group homomorphism is a Lie subgroup, and identifies its Lie algebra.
• Problem 2 gives a criterion for a Lie subgroup to be normal.
• Problem 3 proves the converse of Problem 5 on Problem Sheet E.
• Problem 4 and 5 are about $\mathbb{R}P^n$.
• Problem 6 is about the Klein bottle.
• Problem 7 concerns pullback bundles. (Don’t be scared by this one, it’s really easy.)