Example 18.1 (Some applications to complex analysis):

(i) Let $\Omega$ be a simply connected open subset of $\mathbb{C} - \{0\}$ and $j:\Omega \longrightarrow \mathbb{C} -\{0\}$ be the inclusion and $\exp: \mathbb{C} \longrightarrow \mathbb{C} - \{0\}$ be the exponential map. Then $p$ is a covering projection with respect to which $j$ has a lift ${\tilde j}:\Omega\longrightarrow \mathbb{C}$ which means

$$\exp({\tilde j}(z)) = z,\quad z\in \Omega \eqno(18.3)$$

Thus there is a continuous branch of the logarithm on any simply connected open subset of $\mathbb{C} - \{0\}$. In the exercises the student is asked to show that any continuous lift is holomorphic.

(ii) Consider the map $S: \mathbb{C} - \{0\}\longrightarrow \mathbb{C} - \{0\}$ given by $S(z) = z^2$. Let $\Omega = \mathbb{C} - [0, 1/2]$ and $f:\Omega\longrightarrow \mathbb{C} - \{0\}$ be given by

$$f(z) = z(2z-1). \eqno(16.4)$$

Let us determine the induced map $f_*:\pi_1(\Omega, 1) \longrightarrow \pi_1(\mathbb{C} - \{0\}, 1)$. The group $\pi_1(\Omega, 1)$ is the infinite cyclic group generated by the homotopy class of the loop $\gamma(t) = \exp(2\pi it)$. Since $\mathbb{C} - \{0\}$ is a topological group under multiplication of complex numbers, we may apply corollary (12.2) to get

$$[f\circ\gamma(t)] = [\gamma(t)]+[2\gamma(t)-1]. \eqno(18.5)$$

The additive notation is used for the infinite cyclic group. The last equation may be rewritten as

$$[f\circ\gamma(t)] = [\gamma(t)]+\Big[\gamma(t)\Big(2 - \frac{1}{\... ...ig)\Big] = 2[\gamma(t)] + \Big[2 - \frac{1}{\gamma(t)}\Big] = 2, \eqno(18.6)$$

since $\vert\gamma(t)\vert = 1$ and the loop $\Big(2 - \frac{1}{\gamma(t)}\Big)$ can be contracted to the constant loop in $\mathbb{C} - \{0\}$. Hence

$$f_*(\pi_1(\mathbb{C} - [0, 1/2], 1) = 2\mathbb{Z} = S_*(\mathbb{C} - \{0\}, 1). \eqno(18.7)$$

The lifting criterion holds and $f$ has a unique lift ${\tilde f}$ such that ${\tilde f}(1) = 1$. This lift is the continuous branch of $\sqrt{z(2z-1)}$ defined on $\Omega$. In exercise 3, the student is asked to show that the lift ${\tilde f}$ is holomorphic. Note that the space $\Omega$ is not simply connected.

The next example is Picard's theorem which is a corollary of the following highly non-trivial result.