2017/08/11 by Dayal Dharmasena, Dharmasena, Dayal, Evgeny A. Poletsky +1
Mathematics · #32E30 #Complex Variables (math.CV) #FOS: Mathematics #Primary: 32Q55 #math.CV #msc:32E30 #msc:32H02 #msc:32Q55 #secondary: 32H02
paper · pdf · doi:10.48550/arxiv.1708.04530
arXiv admin note: text overlap with arXiv:1210.1191
arxiv created 2017/08/11 · arxiv updated 2017/08/16
Let W be a domain in a connected complex manifold M and w0∈ W. Let \mathcal Aw0(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, f(∂\mathbb D)⊂ W and f(1)=w0. On the homotopic equivalence classes η1(W,M,w0) of \mathcal Aw0(W,M) we introduce a binary operation ⋆ so that η1(W,M,w0) becomes a semigroup and the natural mappings ι1: η1(W,M,w0)→π1(W,w0) and δ1: η1(W,M,w0)→π2(M,W,w0) are homomorphisms. \par We show that if W is a complement of an analytic variety in M and if S=δ1(η1(W,M,w0)), then S∩ S-1=\e\ and any element a∈π2(M,W,w0) can be represented as a=bc-1=d-1g, where b,c,d,g∈ S. \par Let \mathcal Rw0(W,M) be the space of all continuous mappings of D into M such that f(∂\mathbb D)⊂ W and f(1)=w0. We describe its open dense subset \mathcal R±w0(W,M) such that any connected component of \mathcal R±w0(W,M) contains at most one connected component of \mathcal Aw0(W,M).