The Atiyah–Jones conjecture for rational surfaces

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 question readily generalises 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 k→∞ MIk indeed has the homotopy type of Ck(X). However, understanding the behaviour of the maps tk at finite stages is a finer question. Using Taubes’ results, to prove the Atiyah–Jones conjecture it then suffices to show that the maps tk induce isomorphism in homology and homotopy through a range. In 1993, Boyer, Hurtubise, Milgram and Mann [BHMM] proved that the Atiyah–Jones conjecture holds for the sphere S and in 1995, Hurtubise and Mann [HM] proved that the conjecture is true for ruled surfaces. In this paper I show that if the Atiyah–Jones conjecture holds true for a complex surface X then it also holds for the surface X̃ obtained by blowingup X at a point. In particular, it follows that the conjecture holds true for all rational surfaces.