zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Atiyah, M. F., Instantons in two and four dimensions,Comm. Math. Phys., 93 (1984), 437–451. · Zbl 0564.58040 · doi:10.1007/BF01212288
[2] Dold, A. &Thom, R., Quasifaserung und Unendliche Symmetrische Producte,Ann. of Math., 67 (1958), 239–281. · Zbl 0091.37102 · doi:10.2307/1970005
[3] Donaldson, S. K., Instantons and geometric invariant theory,Comm. Math. Phys., 93 (1984), 453–460. · Zbl 0581.14008 · doi:10.1007/BF01212289
[4] Earle, C. J. &Eells, J., Fibre bundle description of Teichmüller theory,J. Differential Geom., 3 (1969), 19–43. · Zbl 0185.32901
[5] Earle, C. J. &Schatz, A., Teichmüller theory for surfaces with boundary,J. Differential Geom., 4 (1970), 169–185. · Zbl 0194.52802
[6] Farkas, H. M. &Kra, I.,Riemann surfaces. Springer Verlag, New York, 1980. · Zbl 0475.30001
[7] Guest, M. A., Topology of the space of absolute minima of the energy functional,Amer. J. Math., 106 (1984), 21–42. · Zbl 0564.58014 · doi:10.2307/2374428
[8] Hamilton, R. S., The inverse function theorem of Nash and Moser,Bull. Amer. Math. Soc., 7 (1982), 65–222. · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[9] Kirwan, F. C., On spaces of maps from Riemann surfaces to Grassmannians and application to the cohomology of moduli of vector bundles,Ark. Mat., 24 (1986), 221–275. · Zbl 0625.14026 · doi:10.1007/BF02384399
[10] McDuff, D., Configuration spaces of positive and negative particles,Topology, 14 (1975), 91–107. · Zbl 0296.57001 · doi:10.1016/0040-9383(75)90038-5
[11] McDuff, D. &Segal, G., Homology fibrations and the ”Group-completion” theorem.Invent. Math., 31 (1976), 279–284. · Zbl 0312.55021 · doi:10.1007/BF01403148
[12] Pressley, A. &Segal, G.,Loop Groups. Clarendon Press, Oxford, 1986. · Zbl 0618.22011
[13] Seeley, R. T., Extentions ofC functions defined in a half space,Proc. Amer. Math. Soc., 15 (1964), 625–626. · Zbl 0127.28403
[14] Segal, G., The topology of spaces of rational functions,Acta Math., 143 (1979), 39–72. · Zbl 0427.55006 · doi:10.1007/BF02392088
[15] Wells, R. O.,Differential Analysis on Complex Manifolds. Springer Verlag, New York, 1980. · Zbl 0435.32004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.