Exercises:

1. For the map $S$ in example (18.3) show that $S_*$ is the map $\mathbb{Z} \longrightarrow \mathbb{Z}$ given by $x \mapsto 2x$.
2. Suppose $G$ is a path connected topological group with unit element $e$ and $p:{\tilde G} \longrightarrow G$ is a covering map. For any choice of ${\tilde e}\in p^{-1}(e)$ show that there is a group operation on ${\tilde G}$ with unit element ${\tilde e}$ that makes ${\tilde G}$ into a topological group and $p$ is a continuous group homomorphism.
3. Show that if $\Omega$ is an open subset of $\mathbb{C} - \{0\}$ on which a continuous branch of the logarithm exists then this branch is automatically holomorphic. Likewise show that the continuous branch of $\sqrt{z(2z-1)}$ on $\mathbb{C} - [0, 1/2]$ obtained in the lecture is holomorphic.
4. Use the fact that $S^{n-1}$ is not a retract of $S^n$ to prove that $\mathbb{R}P^{n-1}$ is not a retract of $\mathbb{R}P^n$.
5. Show that any continuous map $S^n\longrightarrow S^1$ is homotopic to the constant map if $n\geq 2$. What about maps from the projective spaces $\mathbb{R}P^n\longrightarrow S^1$ ($n\geq 2$)?

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Lecture XIX - Deck Transformations

Given a covering projection $p: \tilde{X} \longrightarrow X$, the deck transformations are, roughly speaking, the symmetries of the covering space. Thus it should not come as a surprise that they play a crucial part in the theory of covering spaces. In this lecture all spaces are assumed to be connected and locally path connected.