Example 15.2:

Consider the exponential map $\exp: \mathbb{C} \longrightarrow \mathbb{C} - \{0\}$ and an open set $\Omega \subset \mathbb{C}$ and the inclusion map

$$j: \Omega \longrightarrow \mathbb{C} - \{0\}.$$

To say that the inclusion map $j$ has a lift with respect to the exponential map means the existence of a continuous ${\tilde j}:\Omega\longrightarrow \mathbb{C}$ such that

$$\exp({\tilde j}(z)) = z,$$   for all $$z \in \Omega.$$

In other words the existence of a lift of $j$ is equivalent to the existence of a continuous branch of the logarithm on $\Omega$. We know from complex variable theory that such a continuous branch need not exist in general such as for instance the case $\Omega = \mathbb{C} - \{0\}$.

In place of the exponential map we could consider the map $S: \mathbb{C} - \{0\}\longrightarrow \mathbb{C} - \{0\}$ given by $S(z) = z^2$. The problem of lifting the inclusion map of a domain $\Omega \subset \mathbb{C} - \{0\}$ is then equivalent to the existence of a continuous branch of the square root function on $\Omega$. We also know from complex analysis that if the lift exists it need not be unique. Well, if a domain $\Omega \subset \mathbb{C} - \{0\}$ admits a continuous branch of the square root then it admits two branches. If it admits a continuous branch of the logarithm then it admits infinitely many any two of which differ by an integer multiple of $2\pi i$. On a connected domain, the branch however is uniquely specified by specifying a value at a point of the domain. The following theorem generalizes this in the context of covering spaces.