Polar Spider Web

The Spider Web Exponential Epimorphism 

The exponential homomorphism \[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \] is clearly an epimorphism, that is, surjective, but it is not injective.

From the complex exponential group epimorphism and the fact that \( \operatorname{\mid exp} z \mid=1 \text { holds if and only if } z \in \mathbb{R} i \) it can be deduced that \( \exp (i \mathbb{R})=S^{1}, \) where \( S^{1} \) denotes the multiplicative circle group. Hence, we obtain the following  polar spider web epimorphism:

The morphism \( p: \mathbb{R} \rightarrow S^{1}, \varphi \mapsto e^{i \varphi} \) is a group epimorphism whose kernel is the group \( 2 \pi \mathbb{Z}, \) and we have: \( p\left(\frac{\pi}{2}\right)=i .\)

Note that for \( \varphi, \psi \in \mathbb{R} \) we have:

\( p(\varphi+\psi)=\exp (i \varphi+i \psi)=(\exp i \varphi)(\exp i \psi)= \) \( p(\varphi) p(\psi) . \)

Thus, \( p \) is an epimorphism, since \( p(\mathbb{R})=\exp (i \mathbb{R})=S^{1} .\) Since Ker(exp) \( =2 \pi i \mathbb{Z} \) we obtain that:

Ker \( p=\{t \in \mathbb{R}: it \in \operatorname{Ker}(\exp )\}= \) \( \{t \in \mathbb{R}: t \in 2 \pi \mathbb{Z}\} . \)