Math Conjectures

The 11/811/8-conjecture

OPENMajorConjectureProposed c. 1980 · Full conjecture

Canonical statement

If XX is a closed, connected, oriented, smooth spin 44-manifold, then
b2(X) ≥ 118 ∣σ(X)∣, b_{2}(X)\ \ge\ \frac{11}{8}\,|\sigma(X)|,
where b2(X)=dim⁡QH2(X;Q)b_{2}(X)=\dim_{\mathbb Q}H_{2}(X;\mathbb Q) and σ(X)\sigma(X) is the signature of the intersection form on H2(X;R)H_{2}(X;\mathbb R).
View source LaTeX
If \(X\) is a closed, connected, oriented, smooth spin \(4\)-manifold, then \[ b_{2}(X)\ \ge\ \frac{11}{8}\,|\sigma(X)|, \] where \(b_{2}(X)=\dim_{\mathbb Q}H_{2}(X;\mathbb Q)\) and \(\sigma(X)\) is the signature of the intersection form on \(H_{2}(X;\mathbb R)\).
GEO-TOP-003:[Furuta2001Monopole118][HopkinsLinShiXu2022IntersectionForms][K3ProblemList2026]
Geometry and topologyMathematical physics

Notes

The 11/811/8-conjecture predicts a sharp constraint on the topology of smooth spin 44-manifolds: a closed connected oriented smooth spin 44-manifold XX should satisfy b2(X)≥118∣σ(X)∣b_2(X)\ge\tfrac{11}{8}|\sigma(X)|, where σ\sigma is the signature; equivalently, if the even unimodular intersection form is 2kE8⊕lH2kE_8\oplus lH, then l≥3∣k∣l\ge3|k|. The bound would be sharp, since the K3K3 surface has b2=22b_2=22 and σ=−16\sigma=-16. The conjecture dates from around 1980 and is usually attributed to Matsumoto, though the first printed modern statement is unclear.

The strongest general result is Furuta's theorem, proved through a Pin(2)\mathrm{Pin}(2)-equivariant analysis of the Seiberg–Witten monopole equations, which gives b2(X)≥54∣σ(X)∣+2b_2(X)\ge\tfrac{5}{4}|\sigma(X)|+2, the 10/8+210/8+2 bound [Furuta2001Monopole118]. Hopkins, Lin, Shi and Xu later sharpened the inequality in certain congruence classes of the signature using the Pin(2)\mathrm{Pin}(2)-equivariant Mahowald invariant [HopkinsLinShiXu2022IntersectionForms].

The gap between the coefficients 10/810/8 and 11/811/8 persists, and the conjecture remains a central open problem in low-dimensional topology [K3ProblemList2026]; closing it would settle which even intersection forms arise from smooth spin 44-manifolds.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.