Acta Mathematica

The topology of rational functions and divisors of surfaces

F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram

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During the preparation of this work each of the authors were supported by NSF grants, the second author by an NSF-PYI award, and the first and fourth authors by the S.F.B. 170 in Göttingen.

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Acta Math. Volume 166 (1991), 163-221.

Received: 6 June 1989
First available in Project Euclid: 31 January 2017

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1991 © Almqvist & Wiksell


Cohen, F. R.; Cohen, R. L.; Mann, B. M.; Milgram, R. J. The topology of rational functions and divisors of surfaces. Acta Math. 166 (1991), 163--221. doi:10.1007/BF02398886.

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  • [And] Andreotti, A., On a theorem of Torelli, Amer. J. Math., 80 (1958), 801–828.
  • [Ar] Artin, E., Theory of braids. Ann. of Math., 48 (1947), 47–72.
  • [AJ] Atiyah, M. F. & Jones, J. D., Topological aspects of Yang-Mills theory, Comm. Math. Phys., 61 (1978), 97–118.
  • [Bi] Birman, J. S., Braids, links and mapping class groups. Ann. of Math. Stud., 82 (1974), Princeton Univ. Press.
  • [Böd] Bödigheimer, C.-F., Gefärbte Konfigurationen: Modelle für die stabile Homotopie von Eilenberg-MacLane-Räumen. Dissertation, University of Heidelberg, 1984.
  • [BCT] Bödigheimer, C.-F., Cohen, F. R. & Taylor, L., On the homology of configuration spaces. To appear in Topology.
  • [BCM] Bödigheimer, C.-F., Cohen, F. R. & Milgram, R. J., On the spaces TPn(Y). To appear.
  • [B] Bogomol'nyi, E. B., The stability of classical solutions. Soviet J. Nuclear Phys., 24 (1976), 449.
  • [BV] Boardman, J. M. & Vogt, R. M., Homotopy-everythingH-spaces. Bull. Amer. Math. Soc., 74 (1968), 1117–1122.
  • [BoMa] Boyer, C. P. & Mann, B. M., Monopoles, non-linear γ models, and two-fold loop spaces. Comm. Math. Phys., 115 (1988), 571–594
  • [Br1] Brockett, R. W., Some geometric questions in the theory of linear systems. IEEE Trans. Automat. Control, 21 (1976), 449–455.
  • [Br2]-—, The geometry of the set of controllable linear systems. Res. Rep. Autom. Control Lab. Nagoya Univ., 24 (1977), 1–7.
  • [BG] Brown, E. H. & Gitler, S., A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Topology, 12 (1973), 283–296.
  • [BP1] Brown, E. H. & Peterson, F. P., On the stable decomposition of Ω2Sr+2. Trans. Amer. Math. Soc., 243 (1978), 287–298.
  • [BP2]—, Relations among characteristic classes II. Ann of Math., 81 (1965), 356–363.
  • [BP3]— A universal space for normal bundles of n-manifolds. Comment. Math. Helv., 54 (1979), 405–430.
  • [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.
  • [Car] Cartan, H., Algebres d'Eilenberg-MacLane et homotopie. Seminaire H. Cartan, Paris, 1954/55.
  • [C] Cohen, F. R., The homology ofCn+1 spaces. Lecture Notes in Mathematics, 533 (1976), 207–352, Springer-Verlag.
  • [CMT] Cohen, F. R., Taylor, L. & May, J. P.. Splitting of certain spaces CX. Math. Proc. Cambridge Philos Soc., 84 (1978) 465–496.
  • [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.
  • [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.
  • [C2M22] Cohen, F. R., Cohen, R. L., Mann, B. M. & Milgram, R. J., The homotopy type of rational functions. To appear in Math. Z.
  • [Co] Cohen, R. L., Odd primary infinite families in stable homotopy theory. Mem. Amer. Math. Soc., 242 (1981).
  • [C2]—, The immersion conjecture for differentiable manifolds. Ann. of Math., 122 (1985), 237–328.
  • [C3] Cohen, R. L., The homotopy theory of immersions. Proc. ICM Warsaw. 1983, 627–639.
  • [Del] Delchamps, D. F., Global structures of families of multivariable systems with an application to identification. Math. Systems Theory, 18 (1985), 329–380.
  • [DT] Dold, A. & Thom, R., Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math., 67 (1958), 239–281.
  • [D] Donaldson, S. K., Nahm's equations and the classification of monopoles. Comm. Math. Phys., 96 (1984), 387–407.
  • [EW1] Eells, J. & Wood, J. C., Restrictions on harmonic maps of surfaces. Topology, 17 (1976), 263–266.
  • [EW2]-—, Harmonic maps from surfaces to complex projective spaces. Adv. in Math., 49 (1983), 217–263.
  • [FaN] Fadell, E. & Neuwirth, L., Configuration spaces. Math. Scand., 10 (1962), 111–118.
  • [FoN] Fox, R. H. & Neuwirth, L., The braid group. Math. Scand., 10 (1962), 119–126.
  • [Fu] Fuks D. B., Cohomologies of the braid groups mod 2. Functional Anal. Appl., 4 (1970), 143–151.
  • [Ga] Ganea, T., A generalization of the homology and homotopy suspension. Comment. math. Helv., 39 (1965), 295–322.
  • [G] Guest, M., Topology of the space of absolute minima of the energy functional. Amer. J. Math., 106 (1984), 21–42.
  • [He1] Helmke, U., The topology of a moduli space for linear dynamical systems. Comment. Math. Helv., 60 (1985), 630–655.
  • [He2]-—, Topology of the moduli space for reachable linear dynamical systems: the complex case. Math. Systems Theory, 19 (1986), 155–187.
  • [H1] Hitchin, N. J., Monopoles and geodesics. Comm. Math. Phys., 83 (1982), 579–602.
  • [H2]—, On the construction of monopoles. Comm. Math. Phys., 89 (1983), 145–190.
  • 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.
  • Kailath, T., Linear Systems. Prentice-Hall, 1980.
  • 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.
  • Löffler, P. & Milgram, R. J., The structure of deleted symmetric products. Artin's Braid group and applications, A.M.S. Summer Institutes.
  • Madsen, Ib & Milgram, R. J., On spherical fiber bundles and their PL reductions. New Developments in Topology, Cambridge University Press, 1974, 43–59.
  • Mahowald, M., A new infinite family in2π ${}_{*}^{5}$ . Topology, 16 (1977), 249–256.
  • [MM] Mann, B. M. & Milgram, R. J., The topology of holomorphic maps from the Riemann sphere to complex Grassmann manifolds. To appear in J. Differential Geom.
  • [May] May, J. P., The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics, 271. Springer-Verlag, 1972.
  • [M] Milgram, R. J., The homology of symmetric products. Trans. Amer. Math. Soc., 138 (1969), 251–265.
  • [M2]—, Iterated loop spaces. Ann. of Math., 84 (1966), 386–403.
  • [Mil] Milnor, J., On spaces having the homotopy type of a CW-complex. Trans. Amer. Math. Soc., 90 (1959), 272–280.
  • [N] Nahm, W., The algebraic geometry of multimonopoles. Lecture Notes in Physics, 180. Springer-Verlag, 1983, pp. 456–466.
  • [Sc] Scott, G. P., Braid groups and the group of homeomorphisms of a surface. Math. Proc. Camb. Philos. Soc., 68 (1970), 605–617.
  • [Seg] Segal, G., The topology of rational functions. Acta Math., 143 (1979), 39–72.
  • [Seg2]—, Configuration spaces and iterated loop spaces. Invent. Math., 21 (1973), 213–221.
  • [Sn] Snaith, V. P., A stable decomposition of ΩnΣnT. J. London Math. Soc (2), 7 (1974), 577–583.
  • [T1] Taubes, C. H., The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on R3; Part I, Comm. Math. Phys., 86 (1982), 257–298; Part II, Comm. Math. Phys., 86 (1982), 299.
  • [T2]—, Monopoles and maps from S2 to S2; the topology of the configuration space. Comm. Math. Phys., 95 (1984), 345–391.
  • [W] Woo, G., Pseudo-particle configurations in two-dimensional ferromagnets. J. Math. Phys., 18 (1977), 1264.