# Kill the Killing form

Let $\mathfrak g$ be a Lie algebra and $K$ its Killing form. Is the kernel of $K$ always equal to the radical of $\mathfrak g$?

# Centers

Let $G$ be a connected Lie subgroup of $\mathrm{GL}_n$. Then $A\in G$ commutes with all of $G$ iff it commutes with all of $\mathfrak{g}$.

# Bounding the projective plane 2

Let $n$ be a natural number. Give an example of a compact $2n$-manifold $M$ such that $\partial M = \mathbb{R}P^{2n-1}$.

# Bounding the projective plane

Is there a compact 3-manifold having $\Bbb{R}\mathrm{P}^2$ as boundary?

# (Not) on a plane

A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.

# No Eckmann-Hilton

Prove the following two statements and conclude that the fundamental group of a (connected) topological group $G$ is abelian:

1) A discrete normal subgroup of a connected topological group is central.
2) If $p:\hat{G}\to G$ is the universal covering, the total space $\hat{G}$ admits a topological group structure such that $p$ is a group morphism. The kernel of $p$ is then isomorphic to the fundamental group of $G$.

# More of not everything being abstract nonsense

The exponential map is surjective for $\mathrm{GL}_n(\mathbb{C})$ but not for $\mathrm{SL}_2(\mathbb{C})$.