### 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 |

### Fingerprint

### Keywords

- Holomorphic mappings
- Homotopic Oka principle
- Homotopy theory

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

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

**Fundamental group and analytic disks.** / Dharmasena, Dayal; Poletsky, Evgeny Alexander.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 371, no. 1, pp. 709-728. https://doi.org/10.1090/tran/7323

}

TY - JOUR

T1 - Fundamental group and analytic disks

AU - Dharmasena, Dayal

AU - Poletsky, Evgeny Alexander

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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).

AB - 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).

KW - Holomorphic mappings

KW - Homotopic Oka principle

KW - Homotopy theory

UR - http://www.scopus.com/inward/record.url?scp=85062099777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062099777&partnerID=8YFLogxK

U2 - 10.1090/tran/7323

DO - 10.1090/tran/7323

M3 - Article

AN - SCOPUS:85062099777

VL - 371

SP - 709

EP - 728

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -