TY - JOUR
T1 - Fundamental group and analytic disks
AU - Dharmasena, Dayal
AU - Poletsky, Evgeny A.
N1 - Funding Information:
Received by the editors August 15, 2016, and, in revised form, May 22, 2017. 2010 Mathematics Subject Classification. Primary 32Q55; Secondary 32H02, 32E30. Key words and phrases. Holomorphic mappings, homotopic Oka principle, homotopy theory. The second author was partially supported by a grant from the Simons Foundation.
Publisher Copyright:
© 2018 American Mathematical Society.
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
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U2 - 10.1090/tran/7323
DO - 10.1090/tran/7323
M3 - Article
AN - SCOPUS:85062099777
SN - 0002-9947
VL - 371
SP - 709
EP - 728
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 1
ER -