Holomorphic fundamental semigroup of Riemann domains

TL;DR

This paper introduces a new semigroup structure called the holomorphic fundamental semigroup for Riemann domains, extending classical fundamental group concepts with holomorphic and homotopy considerations, and provides explicit descriptions in specific cases.

Contribution

It defines the holomorphic fundamental semigroup for Riemann domains, explores its properties, and characterizes it explicitly for finitely connected planar domains.

Findings

The semigroup structure extends the fundamental group with holomorphic properties.

In finitely connected planar domains, the semigroup is fully described.

Equality of classes corresponds to equality of their images in the fundamental group.

Abstract

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ are called {\it $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into $\rS_{1,w_0}({\bar{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. In particular, we completely describe $\eta_1(W,M,w_0)$ when $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is an identity. In particular, we show for a general domain $W\subset\mahbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$.