Holomorphic Fundamental Semigroup of Riemann Domains

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Doctor of Philosophy (PhD)




Eugene Poletsky

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Let (W, Pi) be a Riemann domain over a complex manifold M and w0 be a point in W. Let D be the unit disk in the complex plane and T be the unit circle. Consider the space S1,w0 (D,W,M) of continuous mappings f of T into W such that f(1)=w0 and Pi circ f extends to a holomorphic mapping hat f on D. Mappings f0, f1 in S1,w0 (D, W, M) are called holomorphically homotopic or h-homotopic if there is a continuous mapping ft of [0,1] into S1,w0(D, W, M). Clearly, the h-homotopy is an equivalence relation and the equivalence class of f in S1,w0(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 iota1: η1(W, M, w0) to pi1(W, w0) generated by assigning to f in S1,w0(D, W, M) its restriction to 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 and show that η1(W, M, w0) is an algebraic biholomorphic invariant of Riemann domains. Moreover, iota1([f1] * [f2]) = iota1 ([f1]) cdot iota1 ([f2]), where cdot is the standard operation on pi1(W, w0). Then we establish standard properties of η1(W, M, w0) and provide some examples. When W is a finitely connected domain in the complex plane and Pi is the identity, we show that iota1 is an isomorphism of η1(W, M, w0) onto the minimal subsemigroup of pi1(W, w0) containing some specific generators and invariant with respect to the inner automorphisms. For a general domain W in the complex plane we prove that [f1]=[f2] if and only if iota1([f1]) = iota_1([f2]) which is the manifestation of the homotopic Oka principle.

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