Math 2.0

Manifolds and cell complexes Four-manifolds Conjecture

11/8 conjecture

Builds on 4-manifoldSmooth manifoldSpin structureIntersection formSignature of a quadratic form

Statement

Let MM be a closed, oriented, smooth 4-manifold that admits a spin structure. Let b2(M)b_2(M) be the rank of H2(M;ℤ)H_2(M; \Z) and σ(M)\sigma(M) the signature of its intersection form. Then b2(M)≥118∣σ(M)∣.b_2(M) \ge \frac{11}{8} \abs{\sigma(M)}.

Remarks

The conjecture is open. The K3 surface has b2=22b_2 = 22 and σ=−16\sigma = -16, so it attains equality. For a simply connected MM, the intersection form is even, and by Donaldson's theorem it is indefinite unless it is zero. So it is a sum of copies of ±E8\pm E_8 and of the hyperbolic plane. The conjecture then says that each pair of E8E_8 summands needs at least three hyperbolic summands, as in the K3 surface. Mikio Furuta proved the 10/810/8 theorem: if σ(M)≠0\sigma(M) \ne 0, then b2(M)≥108∣σ(M)∣+2b_2(M) \ge \frac{10}{8} \abs{\sigma(M)} + 2. Later work has improved the constant term of this bound. Without smoothness the inequality fails: the E8E_8 manifold is a closed topological manifold with an even form, b2=8b_2 = 8 and σ=8\sigma = 8.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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4-manifold, concept, no evidence level4-manifoldconceptIntersection form, concept, no evidence levelIntersection formconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSpin structure, concept, no evidence levelSpin structureconcept11/8 conjecture, conjecture, no evidence level11/8 conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. 4-manifold concept
  2. Intersection form concept
  3. Signature of a quadratic form concept
  4. Smooth manifold concept
  5. Spin structure concept
  6. 11/8 conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/4-manifold
About this page
Tag90RA6TAT7H
URIm2://90RA6TAT7H
Kindconjecture
WritingW0 stub
MSC57K40, 57R55, 57R15
StatusThis copy of the site has no build data. Run m2 status 90RA6TAT7H.
About4-manifold, Intersection form
Uses4-manifold, Smooth manifold, Spin structure, Intersection form, Signature of a quadratic form
Changed2 November 2026

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