Let be a connected Lie subgroup of . Then commutes with all of iff it commutes with all of .

# Bounding the projective plane 2

Let be a natural number. Give an example of a compact -manifold such that .

# Bounding the projective plane

Is there a compact 3-manifold having 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 is abelian:

1) A discrete normal subgroup of a connected topological group is central.

2) If is the universal covering, the total space admits a topological group structure such that is a group morphism. The kernel of is then isomorphic to the fundamental group of .

# More of not everything being abstract nonsense

The exponential map is surjective for but not for .

# Not everything is (only) abstract nonsense

Prove that is surjective for every integer butÂ is not necessarily surjective.