Donaldson’s Theorem

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


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].


Modulus Space Gauge Transformation Implicit Function Theorem Chern Number Instanton Modulus Space 
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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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