| Autor: |
Gregory L. Naber |
| Rok vydání: |
2010 |
| Předmět: |
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| Zdroj: |
Texts in Applied Mathematics ISBN: 9781441972538 |
| DOI: |
10.1007/978-1-4419-7254-5_8 |
| Popis: |
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]. |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
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