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On the topology of spaces of holomorphic maps. (English) Zbl 0696.58014
Let Hol(X,Y) and Map(X,Y) be two spaces of holomorphic and continuous maps \(X\to Y\), equipped with the compact-open topology, where X and Y are two complex manifolds. Let further \(Hol^*_ n(X,Y)\) and \(Map^*_ n(X,Y)\) denote spaces of based maps of degree n, and let \({\mathcal V}_ n(X\times CP^ 2,X\vee CP^ 1,G_ C)\) be the space of based isomorphism classes of holomorphic \(G_ C\)-bundles over \(X\times CP^ 1\), trivial over the axis \(X\vee CP^ 1\) and with characteristic class n.
The basic result of the paper is Theorem 7.8. Let X be a Riemann surface and Y a generalized flag manifold or a loop group. If \(X=CP^ 1\), then \[ H_*(Map^*_ 0(CP^ 1,Y))=\lim_{n\to \infty} H_*(Hol^*_ n(CP^ 1,Y)), \] and if \(Y=\Omega G\), then \[ H_*(Map^*_ 0(X,\Omega G))=\lim_{n\to \infty} H_*({\mathcal V}_ n(X\times CP^ 1,X\vee CP^ 1,G_ C)). \]
Reviewer: N.I.Skiba

MSC:
58D15 Manifolds of mappings
54C35 Function spaces in general topology
54C25 Embedding
30H05 Spaces of bounded analytic functions of one complex variable
32H99 Holomorphic mappings and correspondences
14M99 Special varieties
55N99 Homology and cohomology theories in algebraic topology
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