Exercises

1. Suppose that $G$ and ${\tilde G}$ are topological groups and $p:{\tilde G} \longrightarrow G$ is a covering projection that is also a group homomorphism then ker$\;p =$   Deck$({\tilde G}, G)$.
2. Determine the deck transformations for the covering
   $$\sin : \mathbb{C} - \big\{\frac{\pi}{2} + k\pi : k\in \mathbb{Z}\big\} \longrightarrow \mathbb{C} -\{\pm 1\}$$

3. Determine the deck transformations for the covering
   $$p:\mathbb{C} - \big\{\pm 1, \pm 2\big\} \longrightarrow \mathbb{C} -\{\pm 2\}$$
   given by $p(z) = z^3 - 3z$. Show that this covering is not regular. Hint: Use Riemann's removable singularities theorem to show that a deck transformation must be analytic on the whole plane.
4. If $p$ is a prime, what can you say about the group of deck transformations of a $p$-sheeted covering space?
5. Show using the universal property that the universal covering, if it exists is unique upto isomorphism of covering projections.

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Lecture XX - Orbit Spaces

Many interesting spaces in geometry arise as the space of orbits under the action of groups. We have seen examples of this already in lecture 4. An important special case is when the group action is discrete such as the case of the multiplicative group $\{\pm 1\}$ on the sphere $S^n$ resulting in the real projective space $\mathbb{R}P^n$.