Topology, Geometry and Gauge fields pp 403-420 | Cite as

# Donaldson’s Theorem

## Abstract

The moduli space \(\mathcal{M}\) of anti-self-dual connections on the Hopf bundle SU(2) → S^{7} → S^{4} 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 B^{5} 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 S^{4}. Adjoining these points at the ends of the radial segments gives the closed 5-dimensional disc D^{5} which we view as a compactification of the moduli space in which the boundary is a copy of the base manifold S^{4}. In this way the topologies of \(\mathcal{M}\) and S^{4} 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## References

- Don.Donaldson, S.K., “An application of gauge theory to four dimensional topology”, J. Diff. Geo., 18(1983), 279–315.MathSciNetMATHGoogle Scholar
- FU.Freed, D.S., and K.K. Uhlenbeck,
*Instantons and Four-Manifolds*, MSRI Publications, Springer-Verlag, New York, Berlin, 1984.MATHCrossRefGoogle Scholar - Law.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 - Wh.Whitehead, J.H.C., “On simply connected 4-dimensional polyhedra”, Comment. Math. Helv., 22(1949), 48–92.MathSciNetMATHCrossRefGoogle Scholar
- Fr.Freedman, M., “The topology of four-dimensional manifolds”, J. Diff. Geo., 17(1982), 357–454.MATHGoogle Scholar
- Tau1.Taubes, C.H., “Self-dual connections on non-self-dual 4-manifolds”, J. Diff. Geo., 17(1982), 139–170.MathSciNetMATHGoogle Scholar
- DK.Donaldson, S.K., and P.B. Kronheimer,
*The Geometry of Four-Manifolds*, Clarendon Press, Oxford, 1991.Google Scholar - N4.Naber, G.L.,
*Topology, Geometry and Gauge Fields: Interactions*, Springer-Verlag, New York, Berlin, to appear.Google Scholar