# 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 $G$.

# Rank isn’t like dimension

Let $F_2(a,b)$ be the free group in the generators $a$ and $b$. Consider the group morphism $f: F_2(a,b) \to \mathbb{Z}$ defined by $f(a)=f(b)=1$. Prove that $Ker(f)$ is a free group of infinite rank. (Hint: Think topologically!)

# Spooky geometry V

The complement of an algebraic set in $A^n(\Bbb{C})$ is path-connected.

# Contractible manifolds

Classify the closed contractible manifolds (Recall that a manifold is closed if it is compact and its boundary is empty.)

# Connect the dots

Construct a countable, connected topological space with at least two points, satisfying the highest separability axiom you can.

Notice that if two points can be separated by a continuous function, then a connected space with more than one point is uncountable, so your space cannot be $T_{3 \frac 12}$ (or higher).

# No swapping

There is no continuous function $f:\Bbb{R}\to\Bbb{R}$ such that $f(\Bbb{Q})\subseteq \Bbb{R}\setminus\Bbb{Q}$ and $f(\Bbb{R}\setminus\Bbb{Q}) \subseteq \Bbb{Q}$.

# Fixed points of the ball

Brouwer’s theorem shows that any map of the unit ball to itself has a fixed point. Evidently the collection of fixed points is a closed subset of the ball. What closed subsets are realized as the fixed point set of a continuous map of the ball into itself?