Corollary

The mapping $z\mapsto e^z$ of $\mathbf{C}$ into $\mathbf{C}^*$ makes $(\mathbf{C},0)$ a universal covering of $(\mathbf{C}^*,1)$. The group $\pi_1(\mathbf{C}^*,1)$ is isomorphic to $\mathbf{Z}$; the class of the loop $t\mapsto e^{2\pi it}$ is a generator of it. For every integer $n >0$, the mapping $z\mapsto z^n$ makes $\mathbf{C}^*$ a Galois covering of group $\mathbf{Z}/n\mathbf{Z}$ of $\mathbf{C}^*$. Every connected nonempty covering of $\mathbf{C}^*$ is isomorphic to one of these coverings.

The mapping $z\mapsto (|z|, z/|z|)$ is a homeomorphism of the space $\mathbf{C}^*$ onto the space $\mathbf{R}^*_+\times \mathbf{S}^1$ (TG, VI, p. 10, prop. 3); in particular, $\mathbf{C}^*$ is arcwise connected. Since the mapping $x\mapsto e^x$ of $\mathbf{R}$ into $\mathbf{R}_+^*$ is a homeomorphism, it follows from proposition 6 that the mapping $(x, y)\mapsto e^xe^{2\pi iy}$ makes $\mathbf{R}^2$ a covering of $\mathbf{C}^*$. On identifying $\mathbf{R}^2$ with $\mathbf{C}$ by means of the mapping $(x, y)\mapsto x+iy$, we conclude that the space $\mathbf{C}$, endowed with the mapping $z\mapsto e^z$, is a covering of $\mathbf{C}^*$. Since the space $\mathbf{C}$ is connected and simply arcwise connected, the pointed covering $(\mathbf{C},0)$ is a universal covering of the pointed space $(\mathbf{C}^*,1)$.

It also follows from the corollary, III, p. 297, that the Poincaré group of $\mathbf{C}^*$ at 1 is isomorphic to $\mathbf{Z}$ and that the class $\gamma$ of the loop $t\mapsto e^{2\pi it}$ is a generator of it.

Let $n$ be an integer $>0$. The mapping $x\mapsto x^n$ of $\mathbf{R}^*_+$ onto itself is a homeomorphism; it follows as above that the mapping $z\mapsto z^n$ makes $\mathbf{C}^*$ a covering of degree $n$ of $\mathbf{C}^*$. It is Galoisian with group $\mathbf{Z}/n\mathbf{Z}$. The last assertion is proved as in the proof of prop. 6.

For every integer $n\geqslant 0$, the torus $(\mathbf{S}_1)^n$ is an unwindable space (IV, p. 341, prop. 1) and its Poincaré group at every point is isomorphic to $\mathbf{Z}^n($III, p. 297, prop. 4).

We shall generalize these results in paragraph 3 devoted to coverings of topological groups.

3) Real projective spaces. --- Let $n$ be an integer $>0$. The spaces $\mathbf{S}_n$ and $\mathbf{P}_n(\mathbf{R})$ are connected, non-empty, and the canonical mapping (TG, VI, p. $13$)$\varphi :\mathbf{S}_n\rightarrow \mathbf{P}_n(\mathbf{R})$ makes $\mathbf{S}_n$ a principal covering of $\mathbf{P}_n(\mathbf{R})$ with group $\{+1,-1\}($I, p. 99, example 1). For $n\geqslant 2$, the sphere $\mathbf{S}_n$ is simply connected (I, p. 127, example 3) and unwindable, hence simply connected by arcs (IV, p. 344, corollary 1). For $n= 1$, the mapping $z\mapsto z^2$ of $\mathbf{S}_1$ into $\mathbf{S}_1$ defines an equivalence relation in $\mathbf{S}_1$ whose classes are the pairs of opposite points of $\mathbf{S}_1$. The mapping $\varphi :\mathbf{S}_1\rightarrow \mathbf{P}_1(\mathbf{R})$ defines by passing to the quotient a continuous bijection $\psi :\mathbf{S}_1\rightarrow \mathbf{P}_1(\mathbf{R})$ such that $\psi (z^2) =\varphi (z)$. Since $\mathbf{S}_1$ is a compact space, the mapping $\psi$ is a homeomorphism (TG, I, p. 63, cor. 2).