Fundamental group and analytic disks

Abstract:

Let $W$ be a domain in a connected complex manifold $M$ and let $w_0\in W$. Let $\mathcal {A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline {\mathbb D}$ into $M$ that are holomorphic on the interior of $\overline {\mathbb D}$, and let $f(\partial \mathbb {D})\subset W$ 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.

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 $b,c,d,g\in S$.

Let $\mathcal {R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline {\mathbb D}$ 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 $\mathcal {A}_{w_0}(W,M)$.