Exercises

1. Describe a path in $S^n$ whose image under the standard map represents the generator of $\pi_1(\mathbb{R}P^n, x_0)$.
2. Let $C_0$ be the unit circle in the complex plane and $\omega_1, \omega_2,\dots, \omega_n$ denote the $n-$th roots of unity and at each of these a circle $C_j$ of small radius touches the unit circle externally. Construct a continuous map $p$ from the union of these $n+1$ circles onto the figure eight loop such that $p$ is a regular covering. Hint: Take one lobe of the figure eight to be the unit circle $C_0$ and define $p(z) = z^n$ for $z \in C_0$. Let $L$ be the other lobe of figure eight touching the lobe $C_0$ at say the point $1$. For each $j$ let $p_j:C_j \longrightarrow L$ be any homeomorphism such that $p_j(\omega_j) = 1$. Use gluing lemma to glue these maps to obtain the desired covering. in
3. For the covering projection of the preceding exercise determine the action of the fundamental group of the base on a fiber assuming that the loops $C_0$ and $L$ (based at $1$) generate the fundamental group.
4. Consider the covering projection of exercise 6, lecture 15. Show by studying the lifts of various loops based at $(1, 1)$ that the covering is regular. We shall see another proof of regularity of this covering in lecture 19.
5. For the covering considered in the preceding exercise, determine the lifts of the loops
   $$\gamma_1 : t\mapsto 1 - \exp(2\pi it),\quad \gamma_2 : t\mapsto -1 + \exp(2\pi it).$$
   Find the lift of $\gamma_1 *\gamma_2 * \gamma_1^{-1} * \gamma_2^{-1}$ and deduce that the fundamental group of the figure eight space is non-abelian.
6. Show that the figure eight loop $(S^1\times\{1\})\cup (\{1\}\times S^1)$ is not a retract of the torus $S^1\times S^1$. Show that the figure eight loop is a deformation retract of the torus minus a point.

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Lecture XVIII - The lifting criterion

We have already discussed the lifting problem and examined its significance in the light of complex analysis. We have seen in connection with the exponential map/squaring map that the existence of a lift of the inclusion map of a domain $\Omega$ into $\mathbb{C} - \{0\}$ is equivalent to the existence of a continuous branch of the logarithm/square-root function on $\Omega$. Thus it is desirable to have a necessary and sufficient condition for the existence of lifts. We prove one such theorem in this lecture which provides an elegant necessary and sufficient condition.