The 11/8 conjecture for simply connected spin 4-manifolds

Prove that every smooth compact simply-connected spin 4-manifold satisfies the inequality b2(M) ≥ 8|τ(M)|, thereby establishing the 11/8 conjecture.

Background

For a smooth compact simply-connected spin 4-manifold M, the Euler characteristic and signature determine the real intersection form, while Rokhlin’s theorem imposes the divisibility condition τ(M) ≡ 0 mod 16. The inequality b2(M) ≥ 8|τ(M)| would imply that the possible spin homeomorphism types are exhausted by S4 and connected sums of copies of K3 and S2 × S2.

The paper notes that Furuta’s theorem establishes only the weaker bound b2(M) ≥ 4|τ(M)| + 2 when b2(M) is nonzero. The stronger 11/8 inequality is later shown in the paper for simply-connected spin 4-manifolds admitting Einstein metrics, but it remains unresolved in general.