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Donaldson’s Theorem

  • Gregory L. Naber
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 25)

Abstract

The moduli space \(\mathcal{M}\) of anti-self-dual connections on the Hopf bundle SU(2) → S7 → S4 is a rather complicated object, but we have constructed a remarkably simple picture of it. We have identified \(\mathcal{M}\) with the open 5-dimensional unit ball B5 in \({\mathbb{R}}^{6}\). The gauge equivalence class [ω] of the natural connection sets at the center. Moving radially out from [ω] one encounters gauge equivalence classes of connections with field strengths that concentrate more and more at a single point of S4. Adjoining these points at the ends of the radial segments gives the closed 5-dimensional disc D5 which we view as a compactification of the moduli space in which the boundary is a copy of the base manifold S4. In this way the topologies of \(\mathcal{M}\) and S4 are inextricably tied together. In these final sections we will attempt a very broad sketch of how Donaldson [Don] generalized this picture to prove an extraordinary theorem about the topology of smooth 4-manifolds. The details are quite beyond the modest means we have at our disposal and even a bare statement of the facts is possible only if we appeal to a substantial menu of results from topology, geometry and analysis that lie in greater depths than those we have plumbed here. What follows then is nothing more than a roadmap. Those intrigued enough to explore the territory in earnest will want to move on next to [FU] and [Law].

Keywords

Modulus Space Gauge Transformation Implicit Function Theorem Chern Number Instanton Modulus Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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    Donaldson, S.K., “An application of gauge theory to four dimensional topology”, J. Diff. Geo., 18(1983), 279–315.MathSciNetMATHGoogle Scholar
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    Freed, D.S., and K.K. Uhlenbeck, Instantons and Four-Manifolds, MSRI Publications, Springer-Verlag, New York, Berlin, 1984.MATHCrossRefGoogle Scholar
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    Lawson, H.B., The Theory of Gauge Fields in Four Dimensions, Regional Conference Series in Mathematics #58, Amer. Math. Soc., Providence, RI, 1985.Google Scholar
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    Whitehead, J.H.C., “On simply connected 4-dimensional polyhedra”, Comment. Math. Helv., 22(1949), 48–92.MathSciNetMATHCrossRefGoogle Scholar
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    Freedman, M., “The topology of four-dimensional manifolds”, J. Diff. Geo., 17(1982), 357–454.MATHGoogle Scholar
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    Taubes, C.H., “Self-dual connections on non-self-dual 4-manifolds”, J. Diff. Geo., 17(1982), 139–170.MathSciNetMATHGoogle Scholar
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    Donaldson, S.K., and P.B. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press, Oxford, 1991.Google Scholar
  8. N4.
    Naber, G.L., Topology, Geometry and Gauge Fields: Interactions, Springer-Verlag, New York, Berlin, to appear.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Gregory L. Naber
    • 1
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA

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