tangent function determines a local homeomorphism

Question

The question ask me to show that tangent function determines a local homeomorphism $\tan: \mathbb C \to \mathbb C P^1$. I don't understand what the question asking, is the question asking me to show that $\tan:\mathbb C \to \mathbb CP^1$ is a local homeomorphism?

Answer 1

The question is asking you to show that the tangent function $\operatorname{tan}(z)\colon \mathbb{C}\rightarrow \mathbb{CP}^1$ is a local homeomorphism, I guess.

Let me assume that you defined the tangent function as the holomorphic mapping $\mathbb{C}\rightarrow \mathbb{CP}^1$associated to the meromorphic function $\frac{\operatorname{sin}}{\operatorname{cos}}\colon \mathbb{C}\rightarrow \mathbb{C}$. Here, $\operatorname{sin}(z)=\frac{e^{iz}-e^{-iz}}{2i}$ and $\operatorname{cos}(z)=\frac{e^{iz}+e^{-iz}}{2}$ denote the complex sine and cosine functions, respectively. In other words, the tangent function is the holomorphic mapping associated to the meromorphic function $\mathbb{C}\rightarrow\mathbb{C}; z \mapsto\frac{e^{2iz}-1}{i(e^{2iz}+1)}$.

Observe that the tangent function can be written as the composition of a linear automorphism of $\mathbb{C}$, the exponential function $\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}$, and a Möbius transformation. Namely, consider the maps $\phi\colon \mathbb{C}\rightarrow \mathbb{C};z\mapsto 2\pi i z$ and $\psi\colon\mathbb{C}\setminus\{0\}\rightarrow \mathbb{CP}^1; z\mapsto \frac{z-1}{iz+i}$. For all $z\in \mathbb{C}$, we then have $(\psi\circ {\operatorname{exp} }\circ \phi) (z)=\operatorname{tan}(z).$

The exponential function is a local homeomorphism by the inverse function theorem for holomorphic functions. From $\operatorname{det} \big(\begin{smallmatrix} 1 & -1\\ i & i \end{smallmatrix}\big)=2i\neq0$, we see that the linear fractional transformation $\psi$ is bijective, thus a local homeomorphism. Hence, as a composition of local homeomorphisms, the tangent function is a local homeomorphism.

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One can even show that the map $\operatorname{tan}\colon\mathbb{C}\rightarrow \mathbb{CP}^1\setminus\{\pm i\}$ is a covering map. One way to do do is to first convince oneself that the tangent function has the curve lifting property and to then use the fact that any local homeomorphism between manifolds which additionally posseses the curve lifting property is a covering map (see Theorem 4.19 in Forster's Lectures on Riemann Surfaces).