The homotopy type of hyperbolic monopole orbit spaces
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- by Ursula Gritsch PDF
- Proc. Amer. Math. Soc. 128 (2000), 3453-3460 Request permission
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 $\Omega ^{2} (S^{2})$.References
- 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
- 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, DOI 10.1098/rsta.1983.0017
- Peter J. Braam, Magnetic monopoles on three-manifolds, J. Differential Geom. 30 (1989), no. 2, 425–464. MR 1010167
- 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
- Gritsch, U.: Morse theory for the Yang-Mills functional via equivariant homotopy theory, 1997, preprint.
- Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452, DOI 10.1007/BF02684593
- Clifford Henry Taubes, Monopoles and maps from $S^2$ to $S^2$; the topology of the configuration space, Comm. Math. Phys. 95 (1984), no. 3, 345–391. MR 765274, DOI 10.1007/BF01212403
- Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 250324, DOI 10.1016/0040-9383(69)90005-6
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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3453-3460
- MSC (1991): Primary 58B05, 55P91
- DOI: https://doi.org/10.1090/S0002-9939-00-05416-2
- MathSciNet review: 1676340