×

zbMATH — the first resource for mathematics

The topology of rational functions and divisors of surfaces. (English) Zbl 0741.55005
From the authors’ introduction: “Spaces of holomorphic maps between complex manifolds have played a fundamental role in such diverse branches of mathematics as analysis, differential geometry, topology, mathematical physics, and linear control theory. In seminal work, G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] studied the homotopy types of the spaces of holomorphic functions of the 2-sphere \(S^ 2\), of closed surfaces of higher genus, and of spaces of divisors of these surfaces. In particular he showed that the space of holomorphic functions of degree \(k\) fills out the homotopy type of an appropriate function space in a stable range of dimensions (roughly up to dimension \(k-2g\), where \(g\) is the genus). In this paper we continue Segal’s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces.”
The main result of the paper concerns the homotopy type of the space \(Rat_ k\), the space of based holomorphic self-maps of the Riemann sphere having degree \(k\). Segal determined the homotopy type of \(Rat_ k\) through dimension \(k\) [loc. cit.]. This paper shows that \(Rat_ k\) has the stable homotopy type of the Eilenberg-MacLane space \(K(\beta_ 2k,1)\), where \(\beta_ n\) is Artin’s braid group on \(n\)-strings. Further, this stable type is explicitly described in terms of well-known constructions. Connections with spaces of divisors and with moduli spaces of monopoles are also given.
Reviewer: P.J.Kahn (Ithaca)

MSC:
55P42 Stable homotopy theory, spectra
55P99 Homotopy theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [And]Andreotti, A., On a theorem of Torelli,Amer. J. Math., 80 (1958), 801–828. · Zbl 0084.17304 · doi:10.2307/2372835
[2] [Ar]Artin, E., Theory of braids.Ann. of Math., 48 (1947), 47–72. · Zbl 0030.17703
[3] [AJ]Atiyah, M. F. &Jones, J. D., Topological aspects of Yang-Mills theory,Comm. Math. Phys., 61 (1978), 97–118. · Zbl 0387.55009 · doi:10.1007/BF01609489
[4] [Bi]Birman, J. S.,Braids, links and mapping class groups. Ann. of Math. Stud., 82 (1974), Princeton Univ. Press.
[5] [Böd]Bödigheimer, C.-F., Gefärbte Konfigurationen: Modelle für die stabile Homotopie von Eilenberg-MacLane-Räumen. Dissertation, University of Heidelberg, 1984.
[6] [BCT]Bödigheimer, C.-F., Cohen, F. R. & Taylor, L., On the homology of configuration spaces. To appear inTopology. · Zbl 0689.55012
[7] [BCM]Bödigheimer, C.-F., Cohen, F. R. & Milgram, R. J., On the spacesTP n (Y). To appear.
[8] [B]Bogomol’nyi, E. B., The stability of classical solutions.Soviet J. Nuclear Phys., 24 (1976), 449.
[9] [BV]Boardman, J. M. &Vogt, R. M., Homotopy-everythingH-spaces.Bull. Amer. Math. Soc., 74 (1968), 1117–1122. · Zbl 0165.26204 · doi:10.1090/S0002-9904-1968-12070-1
[10] [BoMa]Boyer, C. P. &Mann, B. M., Monopoles, non-linear \(\gamma\) models, and two-fold loop spaces.Comm. Math. Phys., 115 (1988), 571–594 · Zbl 0656.58049 · doi:10.1007/BF01224128
[11] [Br1]Brockett, R. W., Some geometric questions in the theory of linear systems.IEEE Trans. Automat. Control, 21 (1976), 449–455. · Zbl 0332.93040 · doi:10.1109/TAC.1976.1101301
[12] [Br2]—, The geometry of the set of controllable linear systems.Res. Rep. Autom. Control Lab. Nagoya Univ., 24 (1977), 1–7.
[13] [BG]Brown, E. H. &Gitler, S., A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra.Topology, 12 (1973), 283–296. · Zbl 0266.55012 · doi:10.1016/0040-9383(73)90014-1
[14] [BP1]Brown, E. H. &Peterson, F. P., On the stable decomposition of \(\Omega\)2 S r+2.Trans. Amer. Math. Soc., 243 (1978), 287–298. · Zbl 0404.55003 · doi:10.1090/S0002-9947-1978-0500933-4
[15] [BP2]–, Relations among characteristic classes II.Ann of Math., 81 (1965), 356–363. · Zbl 0137.42801 · doi:10.2307/1970620
[16] [BP3]– A universal space for normal bundles ofn-manifolds.Comment. Math. Helv., 54 (1979), 405–430. · Zbl 0415.55011 · doi:10.1007/BF02566284
[17] [BD]Byrnes, C. I. &Duncan, T., On certain topological invariants arising in system theory.New Directions in Applied Math., Springer-Verlag, New York, 1981, pp. 29–71.
[18] [Car]Cartan, H., Algebres d’Eilenberg-MacLane et homotopie.Seminaire H. Cartan, Paris, 1954/55.
[19] [C]Cohen, F. R., The homology ofC n+1 spaces.Lecture Notes in Mathematics, 533 (1976), 207–352, Springer-Verlag.
[20] [CMT]Cohen, F. R., Taylor, L. &May, J. P.. Splitting of certain spacesCX.Math. Proc. Cambridge Philos Soc., 84 (1978) 465–496. · Zbl 0408.55006 · doi:10.1017/S0305004100055298
[21] [CMM]Cohen, F. R., Mahowald, M. &Milgram, R. J.. The stable decomposition of the double loop space of a sphere.Algebraic and Geometric Topology, Proceedings of Symposia in Pure Mathematics, 32 (2) (1978), 225–228.
[22] [C2M2]Cohen, F. R., Cohen, R. L., Mann, B. M. &Milgram, R. J., Divisors and configurations on a surface.Algebraic Topology, Conf. Math., A.M.S., 96 (1989), 103–108. · Zbl 0686.55012
[23] [C2M22]Cohen, F. R., Cohen, R. L., Mann, B. M. & Milgram, R. J., The homotopy type of rational functions. To appear inMath. Z. · Zbl 0790.55005
[24] [Co]Cohen, R. L.,Odd primary infinite families in stable homotopy theory. Mem. Amer. Math. Soc., 242 (1981). · Zbl 0452.55009
[25] [C2]–, The immersion conjecture for differentiable manifolds.Ann. of Math., 122 (1985), 237–328. · Zbl 0592.57022 · doi:10.2307/1971304
[26] [C3]Cohen, R. L., The homotopy theory of immersions.Proc. ICM Warsaw. 1983, 627–639.
[27] [Del]Delchamps, D. F., Global structures of families of multivariable systems with an application to identification.Math. Systems Theory, 18 (1985), 329–380. · doi:10.1007/BF01699476
[28] [DT]Dold, A. &Thom, R., Quasifaserungen und unendliche symmetrische Produkte.Ann. of Math., 67 (1958), 239–281. · Zbl 0091.37102 · doi:10.2307/1970005
[29] [D]Donaldson, S. K., Nahm’s equations and the classification of monopoles.Comm. Math. Phys., 96 (1984), 387–407. · Zbl 0603.58042 · doi:10.1007/BF01214583
[30] [EW1]Eells, J. &Wood, J. C., Restrictions on harmonic maps of surfaces.Topology, 17 (1976), 263–266. · Zbl 0328.58008 · doi:10.1016/0040-9383(76)90042-2
[31] [EW2]—, Harmonic maps from surfaces to complex projective spaces.Adv. in Math., 49 (1983), 217–263. · Zbl 0528.58007 · doi:10.1016/0001-8708(83)90062-2
[32] [FaN]Fadell, E. &Neuwirth, L., Configuration spaces.Math. Scand., 10 (1962), 111–118. · Zbl 0136.44104
[33] [FoN]Fox, R. H. &Neuwirth, L., The braid group.Math. Scand., 10 (1962), 119–126. · Zbl 0117.41101
[34] [Fu]Fuks D. B., Cohomologies of the braid groups mod 2.Functional Anal. Appl., 4 (1970), 143–151. · Zbl 0222.57031 · doi:10.1007/BF01094491
[35] [Ga]Ganea, T., A generalization of the homology and homotopy suspension.Comment. math. Helv., 39 (1965), 295–322. · Zbl 0142.40702 · doi:10.1007/BF02566956
[36] [G]Guest, M., 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
[37] [He1]Helmke, U., The topology of a moduli space for linear dynamical systems.Comment. Math. Helv., 60 (1985), 630–655. · Zbl 0613.93008 · doi:10.1007/BF02567437
[38] [He2]—, Topology of the moduli space for reachable linear dynamical systems: the complex case.Math. Systems Theory, 19 (1986), 155–187. · Zbl 0624.93016 · doi:10.1007/BF01704912
[39] [H1]Hitchin, N. J., Monopoles and geodesics.Comm. Math. Phys., 83 (1982), 579–602. · Zbl 0502.58017 · doi:10.1007/BF01208717
[40] [H2]–, On the construction of monopoles.Comm. Math. Phys., 89 (1983), 145–190. · Zbl 0517.58014 · doi:10.1007/BF01211826
[41] Herman, R. &Martin, C. F., Applications of algebraic geometry to systems theory: The McMillan degree and Kronecker indices of transfer functions as topological and holomorphic systems invariants.SIAM J. Control. Optim., 16 (1978), 743–755. · Zbl 0401.93020 · doi:10.1137/0316050
[42] Kailath, T.,Linear Systems. Prentice-Hall, 1980. · Zbl 0454.93001
[43] Kirwan, F. C., On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles.Ark. Mat., 24 (1986), 221–275. · Zbl 0625.14026 · doi:10.1007/BF02384399
[44] Löffler, P. & Milgram, R. J., The structure of deleted symmetric products.Artin’s Braid group and applications, A.M.S. Summer Institutes.
[45] Madsen, Ib & Milgram, R. J., On spherical fiber bundles and theirPL reductions.New Developments in Topology, Cambridge University Press, 1974, 43–59. · Zbl 0284.55025
[46] Mahowald, M., A new infinite family in2\(\pi\) * 5 .Topology, 16 (1977), 249–256. · Zbl 0357.55020 · doi:10.1016/0040-9383(77)90005-2
[47] [MM]Mann, B. M. & Milgram, R. J., The topology of holomorphic maps from the Riemann sphere to complex Grassmann manifolds. To appear inJ. Differential Geom.
[48] [May]May, J. P.,The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics, 271. Springer-Verlag, 1972. · Zbl 0244.55009
[49] [M]Milgram, R. J., The homology of symmetric products.Trans. Amer. Math. Soc., 138 (1969), 251–265. · Zbl 0177.51404 · doi:10.1090/S0002-9947-1969-0242149-X
[50] [M2]–, Iterated loop spaces.Ann. of Math., 84 (1966), 386–403. · Zbl 0145.19901 · doi:10.2307/1970453
[51] [Mil]Milnor, J., On spaces having the homotopy type of aCW-complex.Trans. Amer. Math. Soc., 90 (1959), 272–280. · Zbl 0084.39002
[52] [N]Nahm, W., The algebraic geometry of multimonopoles.Lecture Notes in Physics, 180. Springer-Verlag, 1983, pp. 456–466.
[53] [Sc]Scott, G. P., Braid groups and the group of homeomorphisms of a surface.Math. Proc. Camb. Philos. Soc., 68 (1970), 605–617. · Zbl 0203.56302 · doi:10.1017/S0305004100076593
[54] [Seg]Segal, G., The topology of rational functions.Acta Math., 143 (1979), 39–72. · Zbl 0427.55006 · doi:10.1007/BF02392088
[55] [Seg2]–, Configuration spaces and iterated loop spaces.Invent. Math., 21 (1973), 213–221. · Zbl 0267.55020 · doi:10.1007/BF01390197
[56] [Sn]Snaith, V. P., A stable decomposition of \(\Omega\) n \(\Sigma\) n T.J. London Math. Soc (2), 7 (1974), 577–583. · Zbl 0275.55019 · doi:10.1112/jlms/s2-7.4.577
[57] [T1]Taubes, C. H., The existence of a non-minimal solution to theSU(2) Yang-Mills-Higgs equations onR 3; Part I,Comm. Math. Phys., 86 (1982), 257–298; Part II,Comm. Math. Phys., 86 (1982), 299. · Zbl 0514.58016 · doi:10.1007/BF01206014
[58] [T2]–, Monopoles and maps fromS 2 toS 2; the topology of the configuration space.Comm. Math. Phys., 95 (1984), 345–391. · Zbl 0594.58053 · doi:10.1007/BF01212403
[59] [W]Woo, G., Pseudo-particle configurations in two-dimensional ferromagnets.J. Math. Phys., 18 (1977), 1264. · doi:10.1063/1.523400
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.