| 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 C and T=6D . Consider the space S1,w0 &parl0;D , W, M) of continuous mappings f of T into W such that f(1) = w0 and pi ∘ f extends to a holomorphic mapping fˆ on D . Mappings f0, f 1 ∈ S1,w0 &parl0;D , W, M) are called holomorphically homotopic or h-homotopic if there is a continuous mapping ft of [0, 1] into S1,w0 &parl0;D , W, M). Clearly, the h-homotopy is an equivalence relation and the equivalence class of f ∈ S1,w0 &parl0;D , W, M) will be denoted by [ f] and the set of all equivalence classes by eta1( W, M, w0).;There is a natural mapping iota1: eta1( W, M, w0) → pi 1(W, w0) generated by assigning to f ∈ S1,w0 &parl0;D , W, M its restriction to T . We introduce on eta1(W, M, w0) a binary operation ☆ which induces on eta 1(W, M, w0) a structure of a semigroup with unity and show that eta1( W, M, w0) is an algebraic biholomorphic invariant of Riemann domains. Moreover, iota1([ f1] ☆ [f2]) = iota 1([f1]) · iota1([ f2]), where · is the standard operation on pi 1(W, w0). Then we establish standard properties of eta1(W, M, w0) and provide some examples. When W is a finitely connected domain in M = C and pi is the identity, we show that iota1 is an isomorphism of eta1(W, M, w 0) 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 ⊂ C we prove that [f1] = [f 2] if and only if iota1([f1]) = iota1([f2]) which is the manifestation of the homotopic Oka principle. |
