## Abstract

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).

Original language | English (US) |
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Pages (from-to) | 709-728 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

## Keywords

- Holomorphic mappings
- Homotopic Oka principle
- Homotopy theory

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics