Abstract
Spaces of particles have long been studied in homotopy theory, partly for their intrinsic interest but also for their role in describing the structure of loop spaces. Recently the structure of these spaces has been put to good use in understanding several moduli spaces of solutions to variational problems, such as the moduli of holomorphic maps of surfaces into certain complex manifolds, the moduli of instantons, and the Chow varieties. In these notes, we give a detailed description of the particle structures involved in the first two cases, and then explain how well-established results on the topology of particle spaces can be exploited to prove stability theorems for the topology of the moduli spaces, theorems which state that the moduli space approximates in a suitable homotopic sense the topology of the function spaces in which they sit, provided one stabilises with respect to a charge or degree.
During the preparation of this work the author was supported by grants from NSERC and FCAR
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F.J. Almgren, Jr., Homotopy groups of the integral cycle groups, Topology 1 (1962), 257–299.
M.F. Atiyah, J.D. Jones, Topological aspects of Yang-Mills theory, Comm. Math.Phys. 61 (1978), 97–118.
A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser. 182, Harlow, 1989.
R.J. Baston and M.G Eastwood, The Penrose Transform,Oxford University Press,1989.
C.P. Boyer, J. Hurtubise, B.M. Mann, R.J. Milgram, The topology of instanton moduli spaces. I: The Atiyah-Jones conjecture,Ann.of Math 137 (1993), 561–609.
C.P. Boyer, J.C. Hurtubise, B.M. Mann, R.J. Milgram, The topology of the space of rational maps into generalised flag manifolds, Acta Math. 173 (1994), 61–101.
C.P. Boyer, J.C. Hurtubise, B.M. Mann, R.J. Milgram, Stability theorems for spaces of rational curves, in preparation.
C.P. Boyer, B.M. Mann, Homology operations on instantons, J. Differential Geom. 28 (1988), 423–465.
K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.
F.R. Cohen, R.L. Cohen, B.M. Mann, R.J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163–221.
F.R. Cohen, T.J. Lada, J.P. May, The Topology of Iterated Loop Spaces, Lecture Notes in Math. 533, Springer, Berlin, 1976.
F.R. Cohen, M. Mahowald, R.J. Milgram, The stable decomposition of the double loop space of a sphere, in: Algebraic and Geometric Topology, Proc. Sympos. Pure Math. 32(2), Amer. Math. Soc., Providence, RI, 1978, 225–228.
R. Cohen, Stable proofs of stable splittings, Math. Proc. Cambridge Philos. Soc. 88 (1980), 149–151.
T. A. Crawford, Full holomorphic maps from the Riemann sphere to complex projective space, J. Differential Geom. 38 (1993), 161–190.
A. Dold, Decomposition theorems for S(n) complexes,Ann.of Math 75 (1962) 8–16.
S.K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–461.
S.K. Donaldson, Anti-self-dual connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1–26.
S.K. Donaldson, P.B. Kronheimer, The Geometry of Four-Manifolds, Oxford Uni-versity Press, 1990.
J. Eells, L. Lemaire, Another report on harmonic maps, Bull.London Math.Soc. 20 (1988), 385–524.
J. Eells, J. Wood, Maps of minimum energy, J. London Math. Soc. (2) 23 (1981), 303–310.
J. Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989), 247–286.
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Rie-mann, Amer.J. Math. 79 (1957), 121–138.
M.A. Guest, Topology of the space of absolute minima of the energy functional, Amer. J. Math. 106 (1984), 21–42.
M.A. Guest, The topology of the space of rational curves on a toric variety, Acta Math. 174 (1995), 119–145.
J.C. Hurtubise, Instantons and jumping lines, Comm. Math. Phys. 105 (1986), 107–122.
J.C. Hurtubise, Holomorphic maps of a Riemann surface into a flag manifold, J. Differential Geom., 43 (1996), 99–118.
J.C. Hurtubise, R.J. Milgram, The Atiyah-Jones conjecture for ruled surfaces, J. Reine Angew. Math.,to appear.
J. Jost, Two-Dimensional Geometric Variational Problems, Wiley Interscience,New York, 1991.
F.C. Kirwan, 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.
F.C. Kirwan, Geometric invariant theory and the Atiyah-Jones conjecture, in: Proc. Sophus Lie Memorial Conf. ( O.A. Laudal and B. Jahren, eds.), Scandinavian University Press, Oslo, 1994.
M. Kuranishi, New proof for the existence of locally complete families of complex structure, in: Proc. Conf. Complex Analysis, Minneapolis 1964, Springer-Verlag, Berlin, 1965, 142–154.
B. Lawson, Algebraic cycles and homotopy theory, Annals of Math. 129 (1989), 253–291.
B.M. Mann, R.J. Milgram, Some spaces of holomorphic maps to complex Grass-mann manifolds, J. Differential Geom. 33 (1991), 301–324.
D. McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107.
R.J. Milgram, Iterated loop spaces, Ann. of Math. 84 (1966) 386–403.
J.P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Math. 271 Springer-Verlag, Berlin, 1972.
W.H. Meeks and S.T. Yau, Topology of three dimensional manifolds and the embedding problem in minimal surface theory,Ann.of Math 112 (1980), 441–484.
W.H. Meeks and S.T. Yau, Group actions on R3, in: The Smith Conjecture ( J.W. Morgan and H. Bass, eds.), Academic Press, New York, 1984, 167–179.
C.S. Seshadri (ed.), Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982).
G. Segal, The topology of rational functions, Acta Math. 143 (1979), 39–72.
V.P. Snaith, A stable decomposition of ΩnΣn X, J. London Math. Soc. (2) 7 (1974), 577–583.
M. Struwe, Variational Methods, Spinger-Verlag, Berlin, 1990.
R. Schoen and S.T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76.
C.H. Taubes, The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989), 163–230.
C.H. Taubes, A framework for Morse theory for the Yang-Mills functional, Invent. Math. 94 (1988) 327–402.
Y. Tian, The based SU(n)-instanton moduli spaces, Math. Ann. 298 (1994), 117–140.
Y. Tian, The Atiyah-Jones conjecture for the classical groups and Bott periodicity, J. Differential Geom.,to appear.
K.K. Uhlenbeck, S.T. Yau, On the existence of Hermitian Yang-Mills connections and stable vector bundles, Comm. Pure. Appl. Math., Suppl. 39 (1986), 257–293.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Hurtubise, J.C. (1997). Moduli spaces and particle spaces. In: Hurtubise, J., Lalonde, F., Sabidussi, G. (eds) Gauge Theory and Symplectic Geometry. NATO ASI Series, vol 488. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1667-3_4
Download citation
DOI: https://doi.org/10.1007/978-94-017-1667-3_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4830-1
Online ISBN: 978-94-017-1667-3
eBook Packages: Springer Book Archive