Fundamental group and analytic disks

Abstract

Let $W$ be a domain in a connected complex manifold $M$ and $w_0\in W$. Let ${\mathcal{A}}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline{D}$ into $M$ that are holomorphic on the interior of $\overline{D}$, $f(\partial \mathbb{D})\subset W$ and $f(1)=w_0$. On the homotopic equivalence classes ${\eta }_1(W,M,w_0)$ of ${\mathcal{A}}_{w_0}(W,M)$ we introduce a binary operation $\star$ so that ${\eta }_1(W,M,w_0)$ becomes a semigroup and the natural mappings ${\iota }_1:{\eta }_1(W,M,w_0)\to {\pi }_1(W,w_0)$ and ${\delta }_1:{\eta }_1(W,M,w_0)\to {\pi }_2(M,W,w_0)$ are homomorphisms. \par We show that if $W$ is a complement of an analytic variety in $M$ and if $S={\delta }_1({\eta }_1(W,M,w_0))$, then $S\cap S^{-1}=\{e\}$ and any element $a\in {\pi }_2(M,W,w_0)$ can be represented as $a=b{c}^{-1}=d^{-1}g$, where $b,c,d,g\in S$. \par Let ${\mathcal{R}}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline{D}$ into $M$ such that $f(\partial \mathbb{D})\subset W$ and $f(1)=w_0$. We describe its open dense subset ${\mathcal{R}}_{w_0}^{\pm }(W,M)$ such that any connected component of ${\mathcal{R}}_{w_0}^{\pm }(W,M)$ contains at most one connected component of ${\mathcal{A}}_{w_0}(W,M)$.

arXiv admin note: text overlap with arXiv:1210.1191