Every Riemann surface \(S\) is the quotient of a free proper holomorphic action of a group \(G\) on the universal cover \(\tilde S\) of \(S\), so \(S\cong \tilde S/G\) is a biholomorphism. Moreover, \(\tilde S\) is biholomorphic to either
- \({\mathbf{CP}}^1\)
- \({\mathbf{C}}\)
- \({\mathbb{D}}\)
Show that there is no continuous square root function defined on all of \({\mathbf{C}}\).
Suppose \(f(z)^2 = z\). Then \(f\) is a section to the covering map \begin{align*} p: {\mathbf{C}}^{\times}&\to {\mathbf{C}}^{\times}\\ z & \mapsto z^2 ,\end{align*} so \(p\circ f = \operatorname{id}\). Using \(\pi_1({\mathbf{C}}^{\times}) = {\mathbf{Z}}\), the induced maps are \(p_*(1) = 2\) and \(f_*(1) = n\) for some \(n\in {\mathbf{Z}}\). But then \(p_* \circ f_*\) is multiplication by \(2n\), contradicting \(p_* \circ f_* = \operatorname{id}\) by functoriality.