### 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) |
---|---|

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

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*371*(1), 709-728. https://doi.org/10.1090/tran/7323