The theorem of Donaldson, that a smooth closed oriented simply-connected 4-manifold with positive definite intersection form must have in fact a standard diagonal intersection form made use of the moduli space of self- dual SU(2) connections with smallest non-trivial Pontryagin class. This was a 5-manifold with singularities to which could be added a boundary which was a copy of the original 4-manifold. The most difficult part of the proof concerned the collar theorem, whereby the boundary was attached. This involved some very subtle analytic computations.
The paper reviewed here studies the moduli space of SO(3)-connections with \(w_ 2\neq 0\) and Pontryagin class less than 4. This is shown to be a compact manifold with singularities, and arguments paralleling those of Donaldson lead to similar, though not as powerful, results. The advantage, for topologists in particular, is that the details of the collar theorem are avoided, but one of the outcomes is still sufficient to prove the existence of exotic differentiable structures on \(R^ 4\), in conjunction with the work of M. Freedman [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. This is expressed in the following theorem: let \(\theta\) be any positive symmetric unimodular integral form. Then \(E_ 8\oplus \theta\) cannot be realized as the intersection form of any smooth closed 4-manifold M with \(H_ 1(M;{\mathbb{Z}})\) containing no 2- torsion.