(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

this paper is to analyse the behavior of a minimizing sequence u k 2 ff for a conformally invariant functional, e.g. the n-energy E(u) =

A Riemannian metric on a compact four-manifold induces a natural L 2 metric on the corresponding moduli space of (anti-) self-dual connections on a principal G- bundle P . When the bundle structure group G is SU(2) and \Gammac 2 (P ) = k, Groisser, Parker and others have found explicit formulas for the components of the L 2 metric on the moduli space M k when k = 1 and the four-manifold is the sphere S 4 or the complex projective space C P 2 . The moduli space M 1 (S 4 ) is diffeomorphic to the open five-ball, while M 1 (C P 2 ) is diffeomorphic to the open cone over C P 2 : these moduli spaces have finite volume and diameter with respect to the L 2 metric. Donaldson, Groisser, and Parker have conjectured that the moduli space M k has finite volume and diameter with respect to the L 2 metric for any integer k. We consider the case where the four-manifold is the spher...

We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blow-up of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. If P → X is a principal SU(2) bundle over a Riemannian four-manifold X, with c2(P) = k> 0, and A is a connection on P, the Yang-Mills functional Y M(A) = ||FA| | 2 is minimal precisely when the curvature FA is anti-self dual, i.e. FA = −∗FA, in which case A is called an instanton of charge k on X. Let MIk(X) denote the moduli space of framed instantons on X with charge k and let Ck(X) denote the space of all framed gauge equivalence classes of connections on X with charge k. In 1978, Atiyah and Jones [AJ] conjectured that the inclusion MIk(X) → Ck(X) induces an isomorphism in homology and homotopy through a range that grows with k. The original statement of the conjecture was for the case when X is a sphere, but the

We show that if the Atiyah–Jones conjecture holds for a surface X, then it also holds for the blow-up of X at a point. Since the conjecture is known to hold for P 2 and for ruled surfaces, it follows that the conjecture is true for all rational surfaces. Given a 4-manifold X, let MIk(X) denote the moduli space of rank 2 instantons on X with charge k and let Ck(X) denote the space of all gauge equivalence classes of connections on X with charge k. In 1978, Atiyah and Jones [AJ] conjectured that the inclusion MIk(X) → Ck(X) induces an isomorphism in homology and homotopy through a range that grows with k. The original statement of the conjecture was for the case when X is a sphere, but the question readily generalizes for other 4-manifolds. The stable topology of these moduli spaces was understood in 1984, when Taubes [Ta] constructed instanton patching maps tk: MIk(X) → MIk+1(X) and showed that the stable limit lim MIk indeed has the ho-

The Palais-Smale Condition C holds for the Yang-Mills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for Yang-Mills over compact manifolds of dimension ≥ 4. The example of SO(3)-invariant SU(2)-connections over S 4, see [2], [14], and [16], suggested that Condition C holds for Yang-Mills over compact manifolds of any dimension when restricted to connections that are invariant under a group action on the manifold with orbits of codimension ≤ 3. Such a result, essentially Theorem 3 below, was established by T. Parker [14]. In this paper we generalize his result. Let X be a smooth compact Riemannian manifold of any dimension, let G be a compact Lie group, and let P be a smooth principal G-bundle over X. Let A denote the space of smooth connections on P and G the group of smooth bundle automorphisms of P. We denote the gauge equivalence class of A ∈ A by [A] ∈ A/G. We fix a G-invariant positive definite inner product on the Lie algebra g. Then the Yang-Mills functional is given by

Dedicated to Nigel Hitchin on the occasion of his sixtieth birthday. In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on S 1 × R 3) and solutions to Nahm’s equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry. 1