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 $a=bc^{-1}=d^{-1}g$, 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 ${\mathcal R}^{\pm}_{w_0}(W,M)$ such that any connected component of ${\mathcal R}^{\pm}_{w_0}(W,M)$ contains at most one connected component of ${\mathcal A}_{w_0}(W,M)$.