Fundamental group and analytic disks

Let W be a domain in a connected complex manifold M and let w ₀ ∈ W . Let A _w0 (W, M) be the space of all continuous mappings of a closed unit disk D into M that are holomorphic on the interior of D, and let f (∂D) ⊂ W and f (1) = w ₀ . On the homotopic equivalence classes η ₁ (W, M, w ₀ ) of A _w0 (W, M) we introduce a binary operation ⋆ so that η ₁ (W, M, w ₀ ) becomes a semigroup and the natural mappings ι ₁ : η ₁ (W, M, w ₀ ) → π ₁ (W, w ₀ ) and δ ₁ : η ₁ (W, M, w ₀ ) → π ₂ (M, W, w ₀ ) are homomorphisms. We show that if W is a complement of an analytic variety in M and if S = δ ₁ (η ₁ (W, M, w ₀ )), then S ∩ S ⁻¹ = {e} and any element a ∈ π ₂ (M, W, w ₀ ) can be represented as a = bc ⁻¹ = d ⁻¹ g, where b, c, d, g ∈ S. Let R _w0 (W, M) be the space of all continuous mappings of D into M such that f (∂D) ⊂ W and f (1) = w ₀ . We describe its open dense subset R ^± _w0 (W, M) such that any connected component of R ^± _w0 (W, M) contains at most one connected component of A _w0 (W, M).