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The homotopy type of hyperbolic monopole orbit spaces


Author: Ursula Gritsch
Journal: Proc. Amer. Math. Soc. 128 (2000), 3453-3460
MSC (1991): Primary 58B05, 55P91
Published electronically: May 18, 2000
MathSciNet review: 1676340
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Abstract:

We prove that the space ${\mathcal{B}}_{U(1)}^{0}$ of equivalence classes of $U(1)$-invariant connections on some $SU(2)$-principle bundles over $S^{4}$ is weakly homotopy equivalent to a component of the second loop space $\Omega ^{2} (S^{2})$.


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  • [A] M. F. Atiyah, Magnetic monopoles in hyperbolic spaces, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 1–33. MR 893593
  • [AB] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, 10.1098/rsta.1983.0017
  • [Ba] Peter J. Braam, Magnetic monopoles on three-manifolds, J. Differential Geom. 30 (1989), no. 2, 425–464. MR 1010167
  • [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
  • [G] Gritsch, U.: Morse theory for the Yang-Mills functional via equivariant homotopy theory, 1997, preprint.
  • [Se] Graeme Segal, Equivariant 𝐾-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 0234452
  • [T] Clifford Henry Taubes, Monopoles and maps from 𝑆² to 𝑆²; the topology of the configuration space, Comm. Math. Phys. 95 (1984), no. 3, 345–391. MR 765274
  • [Wa] Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 0250324

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Additional Information

Ursula Gritsch
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, United Kingdom
Address at time of publication: Department of Mathematics, University of California at Berkeley, Evans Hall, Berkeley, California 94705
Email: ursula@dpmms.cam.ac.uk, ursula@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05416-2
Keywords: Monopoles, gauge theory, equivariant homotopy theory
Received by editor(s): October 30, 1998
Received by editor(s) in revised form: January 15, 1999
Published electronically: May 18, 2000
Additional Notes: This note is part of the author’s Ph.D. thesis written at Stanford University, 1997. The author thanks her advisor Ralph Cohen for constant support and encouragement and the Studienstifung des deutschen Volkes for a dissertation fellowship. Part of this paper was written while the author was supported by an EPSRC Assistantship
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society