Atiyah – Jones conjecture for blown - up surfaces

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 P2 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 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 k→∞ MIk indeed has the homotopy type of Ck(X). However, understanding the behavior of the maps tk at finite stages is a finer question. Using Taubes’ results, to prove the Atiyah–Jones conjecture it would now suffice 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 we 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. Kobayashi and Hitchin gave a one-to-one correspondence between instantons on a topological bundle E over X and holomorphic structures on E, see [LT]. Using this correspondence we translate the Atiyah–Jones conjecture into the language of holomorphic bundles and compare the topologies of the moduli spaces Mk(X) and Mk(X̃) of stable holomorphic bundles on X, resp. X̃, having c1 = 0 and c2 = k.