#### Title

Holomorphic Fundamental Semigroup of Riemann Domains

#### Date of Award

5-2013

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Advisor(s)

Eugene Poletsky

#### Subject Categories

Mathematics

#### Abstract

Let (*W*, Pi) be a Riemann domain over a complex manifold *M* and *w*_{0} be a point in *W*. Let D be the unit disk in the complex plane and T be the unit circle. Consider the space S_{1,w0} (D,W,M) of continuous mappings* f* of T into *W* such that *f*(1)=*w*_{0} and Pi circ *f* extends to a holomorphic mapping hat *f* on D. Mappings *f*_{0}, *f*_{1} in *S*_{1,w0} (D, W, M) are called *holomorphically homotopic or h-homotopic* if there is a continuous mapping *f _{t}* of [0,1] into

*(D, W, M). Clearly, the h-homotopy is an equivalence relation and the equivalence class of*

**S**1,_{w}0*f*in

*(D, W, M) will be denoted by [*

**S**1,_{w}0*f*] and the set of all equivalence classes by η

_{1}(W, M,

*w*

_{0}).

There is a natural mapping iota_{1}: η_{1}(W, M, *w*_{0}) to pi_{1}(W, *w*_{0}) generated by assigning to* f* in * S1,_{w}0*(D, W, M) its restriction to T. We introduce on η

_{1}(W, M,

*w*

_{0}) a binary operation * which induces on η

_{1}(W, M,

*w*

_{0}) a structure of a semigroup with unity and show that η

_{1}(W, M,

*w*

_{0}) is an algebraic biholomorphic invariant of Riemann domains. Moreover, iota

_{1}([

*f*

_{1}] * [

*f*

_{2}]) = iota

_{1}([

*f*

_{1}]) cdot iota

_{1}([

*f*

_{2}]), where cdot is the standard operation on pi

_{1}(W,

*w*

_{0}). Then we establish standard properties of η

_{1}(W, M,

*w*

_{0}) and provide some examples. When

*W*is a finitely connected domain in the complex plane and Pi is the identity, we show that iota

_{1}is an isomorphism of η

_{1}(W, M,

*w*

_{0}) onto the minimal subsemigroup of pi

_{1}(W,

*w*

_{0}) containing some specific generators and invariant with respect to the inner automorphisms. For a general domain W in the complex plane we prove that [

*f*

_{1}]=[

*f*

_{2}] if and only if iota

_{1}([

*f*

_{1}]) = iota_1([

*f*

_{2}]) which is the manifestation of the homotopic Oka principle.

#### Recommended Citation

Dharmasena, Dayal Buddhika, "Holomorphic Fundamental Semigroup of Riemann Domains" (2013). *Mathematics - Dissertations*. 71.

https://surface.syr.edu/mat_etd/71