2012/10/03 by Dayal Dharmasena, Dharmasena, Dayal, Evgeny A. Poletsky +1 · 1 citation
Mathematics · #32E30 (Secondary) #32Q55 (Primary) 32H02 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #math.CV #msc:32E30 #msc:32H02 #msc:32Q55
paper · pdf · doi:10.48550/arxiv.1210.1191
The paper was drastically reworked and corrected and got the new title
openalex publication_date 2012/10/03 · arxiv created 2017/08/11 · arxiv updated 2017/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let (W,Π) be a Riemann domain over a complex manifold M and w0 be a point in W. Let \mathbb D be the unit disk in \mathbb C and \mathbb T=\bd\mathbb D. Consider the space \mathcal S1,w0(\mathbb D,W,M) of continuous mappings f of \mathbb T into W such that f(1)=w0 and Π∘ f extends to a holomorphic on \mathbb D mapping f. Mappings f0,f1∈\mathcal S1,w0(\mathbb D,W,M) are called \it h-homotopic if there is a continuous mapping ft of [0,1] into \rS1,w0(\mathbb D,W,M). Clearly, the h-homotopy is an equivalence relation and the equivalence class of f∈\mathcal S1,w0(\mathbb D,W,M) will be denoted by [f] and the set of all equivalence classes by η1(W,M,w0). There is a natural mapping ι1: η1(W,M,w0)→π1(W,w0) generated by assigning to f∈\mathcal S1,w0(\mathbb D,W,M) its restriction to \mathbb T. We introduce on η1(W,M,w0) a binary operation ⋆ which induces on η1(W,M,w0) a structure of a semigroup with unity. Moreover, ι1([f1]⋆[f2])=ι1([f1])⋅ι1([f2]), where ⋅ is the standard operation on π1(W,w0). Then we establish standard properties of η1(W,M,w0) and provide some examples. In particular, we completely describe η1(W,M,w0) when W is a finitely connected domain in M=\mathbb C and Π is an identity. In particular, we show for a general domain W⊂\mahbb C that [f1]=[f2] if and only if ι1([f1])=ι1([f2]).