Donaldson's invariants
/ DON-uld-sun /
Two four-dimensional spaces can be the same topologically — homeomorphic, the same up to continuous deformation — yet be genuinely different as smooth manifolds, with no smooth way to identify them. Detecting that subtle difference seemed hopeless until Simon Donaldson, around 1983, found a way to manufacture numbers from physics: count solutions of the Yang-Mills equations on the manifold. These counts, the Donaldson invariants, are unchanged by smooth deformations but can tell smooth structures apart, and they revealed that four dimensions is wilder than any other.
Here is the construction in outline. Fix a smooth closed oriented 4-manifold X with a principal SU(2)- (or SO(3)-) bundle and a Riemannian metric. Look at connections A on this bundle whose curvature F_A is anti-self-dual, *F_A = -F_A — the ASD or instanton equation, the absolute minima of the Yang-Mills energy. Modulo gauge, the space of ASD connections is the moduli space M, a finite-dimensional space whose dimension is computed by an index formula. Donaldson's idea: M carries natural cohomology classes (built from the universal bundle), and you integrate, or pair, these classes against M to get numbers. After dealing with the technical pain — M may be non-compact and singular — these numbers are independent of the metric used (a metric is needed to define ASD, but the answer does not depend on it), so they are smooth invariants of X. Packaged together they form the Donaldson polynomial, a polynomial function on the homology H_2(X).
The payoff was spectacular. Donaldson's diagonalizability theorem constrained which intersection forms smooth 4-manifolds can have, immediately producing topological 4-manifolds (via Freedman's classification) that carry no smooth structure at all, and the invariants distinguish smooth structures on homeomorphic manifolds — the failure of the smooth h-cobordism theorem in dimension four, and the existence of exotic R^4, are downstream of this circle of ideas. The honest caveats are heavy. First, defining the invariants rigorously requires serious gauge-theoretic analysis (compactness via Uhlenbeck, perturbing to achieve transversality, orientations of moduli spaces); the 'count' is not naive. Second, Donaldson theory was largely superseded for computations by Seiberg-Witten theory in 1994, which is vastly easier to handle (abelian, compact moduli) and conjecturally equivalent (Witten's conjecture relating the two); for practical four-manifold problems people now usually reach for Seiberg-Witten. Donaldson invariants remain the historically pivotal and conceptually rich original.
Donaldson's first theorem: if a smooth closed 4-manifold has a positive-definite intersection form, that form must be diagonalizable over the integers (equivalent to the standard form). Freedman had built topological 4-manifolds with non-diagonalizable definite forms (e.g. E8); Donaldson's instanton count shows these admit no smooth structure at all — a topological manifold with no smoothing.
Instanton counts forbid the E8 form on a smooth 4-manifold, producing a topological manifold with no smooth structure.
Donaldson invariants are smooth (diffeomorphism) invariants, not homeomorphism invariants — that is exactly the point: they distinguish smooth structures that the homeomorphism type cannot see, and they require careful compactness and transversality, so the word 'count' hides substantial analysis.