Holomorphic fundamental semigroup of Riemann domains

Abstract

Let $(W,Π)$ 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 $Π\circ f$ 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 $η_1(W,M,w_0)$. There is a natural mapping $ι_1:\,η_1(W,M,w_0)\toπ_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $η_1(W,M,w_0)$ a binary operation $\star$ which induces on $η_1(W,M,w_0)$ a structure of a semigroup with unity. Moreover, $ι_1([f_1]\star[f_2])=ι_1([f_1])\cdotι_1([f_2])$, where $\cdot$ is the standard operation on $π_1(W,w_0)$. Then we establish standard properties of $η_1(W,M,w_0)$ and provide some examples. In particular, we completely describe $η_1(W,M,w_0)$ when $W$ is a finitely connected domain in $M=\mathbb C$ and $Π$ is an identity. In particular, we show for a general domain $W\subset\mahbb C$ that $[f_1]=[f_2]$ if and only if $ι_1([f_1])=ι_1([f_2])$.