Exponentiation

Exponential Group Homomorphism - The \( \pi \)-Tuning

The morphism of groups:
\[\exp : \mathbf{C} \rightarrow \mathbf{C}^{\mathbf{x}}, \quad \boldsymbol{z} \mapsto \exp z \]
is a homomorphism of the additive group \( \mathrm{C} \) into the multiplicative group \( \mathrm{C}^{\times}. \) As in the case of a general group homomorphism \( \sigma: G \rightarrow H \) we should consider the image group \( \sigma(G):=\exp (\mathrm{C}), \) and the kernel:
\[\text { Ker } \sigma:=\{g \in G: \sigma(g)=\text { neutral element of } H\}\]
For the exponential group homomorphism we obtain:
\[\exp (\mathrm{C})=\mathrm{C}^{\times}, \quad \operatorname{Ker}(\mathrm{exp})=2 \pi i \mathbb{Z}.\]
We conclude that there is a uniquely defined real number \( \pi>0 \), such that the numbers \( 2 n \pi i, n \in \mathbb{Z}, \) constitute the set of numbers mapped on to 1 by the exponential mapping exp \( z \); Equivalently there is a unique tuning real number number \( \pi \) with the property that:
\[\{w \in \mathbf{C}: \exp w=1\}=2 \pi i \mathbb{Z}.\]
The above property characterizes \( \pi \) uniquely, thus it amounts to the definition of \( \pi \).

Comments

Post a Comment

Popular posts from this blog

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...
Read more

Integral Cohomology and Geometry

Geometric Realization of Second Integral Cohomology Classes The second integral cohomology classes in \( H^{2}(M ; Z) \) are in natural \( 1-1 \) correspondance with the isomorphism classes of circle-bundles over \( M \), where $M$ is a manifold. In particular, every second integral cohomology class \( \hat{F} \in H^{2}(M ; Z) \) has a geometric realization which consists of a space \( P=P_{F} \) together with an action of the circle \( S^{1} \) on \(P \) and a natural projection: \[\begin{array} \\ P \\ \downarrow \pi \\ M \end{array}\] so that the \( S^{1} \) action on \( P \) is free, preserves \( \pi, \) and \( \pi \) induces an isomorphism of \( P / S^{1} \) with \( M .\) For instance if \( M \) is a Riemann surface, consider that $P$ is the set of its unit tangent vectors to $M$, and let \( \pi \) be the projection map which assigns to each tangent vector the corresponding point of tangency. The circle \( S^{1} \) then acts freely by rotating any vector counter-clockwise in its...
Read more

Free Group

Uniformization  A Schottky group is a group \( \Gamma \) with generators \( \gamma_{1}, \ldots, \gamma_{p}, p \geq 1, \) such that there exist \( 2 p \) disjoint Jordan curves \( l_{1}, l_{1}^{\prime}, \ldots, l_{p}, l_{p}^{\prime} \) bounding a \( 2 p- \) connected region \( D \) for which \( \gamma_{j}(D) \cap D=\varnothing \) and \( \gamma_{j}\left(l_{j}\right)=l_{j}^{\prime}, j=1, \ldots, p . \) The group \( \Gamma \) is necessarily free and purely loxodromic, i.e., all its elements \( \gamma \in \Gamma \backslash\{I\} \) are loxodromic or hyperbolic. The factor \( \Omega(\Gamma) / \Gamma \) is a closed surface of genus \( p . \) The geometric approach to Schottky groups is based on the concept of a fundamental region. A fundamental region of a discontinuous group \( \Gamma \) is defined to be a set \( F \subset \Omega(\Gamma) \) containing one point from each orbit \( \Gamma z_{0}, z_{0} \in \Omega(\Gamma), \) and such that each nonempty component \( F \cap \Omega \) of it ...
Read more