The Technique of Fintushel and Stern

Abstract

We remarked at the end of §2 that to study the moduli space of self-dual connections on a G bundle over M we impose three topological conditions on M and G—we require that the intersection form ω be positive definite, the first Betti number b₁ vanish, and the dimension of G be 3—and that there is real trouble if we relax any of these constraints. The differential topologists Ronald Fintushel and Ronald Stern noticed that for G = SO(3), i.e., for oriented real three dimensional vector bundles, a theorem different from Donaldson’s can be obtained. Their nonsmoothability result holds for compact oriented 4-manifolds with almost any finite fundamental group, but not all intersection pairings are allowed. (However, their proof does apply to E₈ ⊕ E₈, and then the existence of fake ℝ⁴ follows as before.) One advantage to their approach is that the analysis is much easier. Since their results have important ramifications for 3-manifold topology, we include an “easy” case of their theorem in this chapter. The difficulties in harder cases are not in the analysis, but arise mostly from the number theory of the intersection form, and we provide enough information so that the reader can fill in these details.