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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Classifying spaces and Dirac operators coupled to instantons

Author: Marc Sanders
Journal: Trans. Amer. Math. Soc. 347 (1995), 4037-4072
MSC: Primary 58D27; Secondary 55P99, 55R45, 57R57, 58G03
MathSciNet review: 1311915
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Abstract: Let $ M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $ SU(l)$ instantons on principal bundles over $ {S^4}$ with second Chern class equal to $ k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $ l$ increases relative to $ k$. This "coupling" procedure produces maps $ {\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $ l$ such maps recover Kirwan's $ [$K$ ]$ homotopy equivalence $ M(k,SU) \simeq BU(k)$. We also compute, for any $ k$ and $ l$, the image of the homology map $ {({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $ Sp(l)$ instantons.

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  • [A] M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford 19 (1968), 113-140. MR 0228000 (37:3584)
  • [AHS] M.F. Atiyah, N.J. Hitchen, and I.M. Singer, Self-duality in four dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425-461. MR 506229 (80d:53023)
  • [ADHM] M.F. Atiyah, V. G. Drinfeld, N.J. Hitchin, and Y. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), 185-187. MR 598562 (82g:81049)
  • [AJ] M.F. Atiyah and J.D.S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), 97-118. MR 503187 (80j:58021)
  • [BHMM] C.P. Boyer, J.C. Hurtubise, B.M. Mann, and R.J. Milgram, The topology of instanton moduli spaces I: The Atiyah-Jones conjecture, Ann. of Math. 137 (1993), 561-609. MR 1217348 (94h:55010)
  • [BM] C.P. Boyer and B.M. Mann, Homology operations on instantons, J. Differential Geom. 28 (1988), 423-465. MR 965223 (89m:58044)
  • [BMW] C.P. Boyer, B.M. Mann, and D. Waggoner, On the homology of $ SU(n)$ instantons, Trans. Amer. Math. Soc. 323 (1991), 529-561. MR 1034658 (92a:58021)
  • [CJ] R. Cohen and J.D.S. Jones, Monopoles, braid groups, and the Dirac operator, Comm. Math. Phys. 158 (1993), 241-246. MR 1249594 (95d:58126)
  • [CLM] F. R. Cohen, T. J. Lada, and J.P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, 1976. MR 0436146 (55:9096)
  • [D] S.K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453-460. MR 763753 (86m:32043)
  • [DK] S. K. Donaldson and P.B. Kronheimer, The geometry of four-manifolds, Oxford Math. Monographs, Oxford Univ. Press, Oxford, 1990. MR 1079726 (92a:57036)
  • [H] D. Husemoller, Fibre bundles, Springer-Verlag, New York, 1966.
  • [K] F. Kirwan, Geometric invariant theory and the Atiyah-Jones conjecture, Proceedings of the Sophus Lie Memorial Conference (Oslo, 1992), editors, O. A. Laudal and B. Jahren, Scandinavian University Press, 1994, pp. 161-188. MR 1456466 (98e:58036)
  • [Ko] S. Kochman, The homology of the classical groups over the Dyer-Lashof algebra, Bull. Amer. Math. Soc. 77 (1971), 142-147. MR 0314049 (47:2601)
  • [Mu] J.R. Munkres, Elementary differential topology, Ann. of Math. 54, Princeton Univ. Press, Princeton, NJ, 1963. MR 0163320 (29:623)
  • [MT] M. Mimura and H. Toda, Topology of Lie groups, I and II, Transl. Math. Monographs, vol. 91, Amer. Math. Soc., Providence, RI, 1991. MR 1122592 (92h:55001)
  • [OSS] C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston, 1980. MR 561910 (81b:14001)
  • [P] S. Priddy, Dyer-Lashof operations for the classifying spaces of certain matrix groups, Quart. J. Math. Oxford 26 (1975), 179-193. MR 0375309 (51:11505)
  • [Ti] Y. Tian, The based $ SU(n)$-instanton moduli spaces, Math. Ann. 298 (1994), 117-139. MR 1252821 (95d:58026)
  • [T1] C. H. Taubes, Self-dual connections on four-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), 517-560. MR 755237 (86b:53025)
  • [T2] C. H. Taubes, The stable topology of selfdual moduli spaces, J. Differential Geom. 29 (1989), 163-230. MR 978084 (90f:58023)

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Keywords: Instantons, Dirac operators, classifying space
Article copyright: © Copyright 1995 American Mathematical Society

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