Fundamental group and analytic disks

Dayal Dharmasena, Evgeny Alexander Poletsky

Research output: Contribution to journalArticle

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 11 (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 languageEnglish (US)
Pages (from-to)709-728
Number of pages20
JournalTransactions of the American Mathematical Society
Volume371
Issue number1
DOIs
StatePublished - Jan 1 2019

Keywords

  • Holomorphic mappings
  • Homotopic Oka principle
  • Homotopy theory

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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