Holomorphic fundamental semigroup of Riemann domains

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 ${\iota }_1([f_1])={\iota }_1([f_2])$.

The paper was drastically reworked and corrected and got the new title

References in corpus (1)

• Envelopes of holomorphy and holomorphic discs