The 11/8-Conjecture for Closed Smooth Spin 4-Manifolds (Matsumoto's Conjecture)

The 11/8-conjecture, formulated by Matsumoto in 1982, is a central open problem of smooth 4-manifold topology: every closed smooth spin 4-manifold $M$ must satisfy $b_2(M)\ge\frac{11}{8}|\operatorname{sign}(M)|$, where $b_2$ is the second Betti number and $\operatorname{sign}$ is the signature of the intersection form. Equivalently, the even indefinite unimodular lattice $2pE_8\oplus qH$ occurs as the intersection form of such a manifold only if $q\ge 3p$; the converse is realized by connected sums of the K3 surface. The strongest known bound is Furuta's 10/8 theorem ($q\ge 2p+1$), sharpened by Hopkins–Lin–Shi–Xu (2018, published 2022) to a 10/8+4 theorem for $p\ge 2$. The same paper completely solves Furuta's question on the existence of Furuta–Mahowald classes and thereby proves that the stable-homotopy method through the Bauer–Furuta invariant cannot, by itself, improve the bound beyond $\frac{10}{8}+4$. The conjecture remains open: closing the gap from $\frac{10}{8}+4$ to $\frac{11}{8}$ requires new geometric or analytic information about the Seiberg–Witten map beyond its equivariant homotopy class.

Background

For a closed oriented 4-manifold $M$, the intersection form $Q_M$ is the symmetric bilinear form on $H^2(M;\mathbb{Z})$ defined by the cup product, $Q_M(a,b)=\langle a\cup b,[M]\rangle$; it is unimodular (determinant $\pm 1$) by Poincaré duality and is called even when $Q_M(x,x)$ is even for every $x$. Freedman's theorem classifies simply connected topological 4-manifolds by $Q_M$ together with the Kirby–Siebenmann obstruction to smoothability, while Donaldson's diagonalizability theorem forces any definite form realized by a smooth 4-manifold to be diagonal. By the Hasse–Minkowski classification, every indefinite unimodular form is isomorphic either to a diagonal form (odd case), realized by connected sums of $\mathbb{C}P^2$, or to $2pE_8\oplus qH$ (even case), where $E_8$ is the unique even unimodular positive-definite form of rank 8, $H$ is the hyperbolic plane $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$, and $p\ge 0$, $q\ge 0$; Rokhlin's theorem forces $\sigma\equiv 0\pmod{16}$. Wu's formula makes evenness of $Q_M$ the algebraic face of the spin condition ($w_2=0$), so the even case is exactly the geography problem for smooth spin 4-manifolds: which forms $2pE_8\oplus qH$ are realized by closed smooth spin 4-manifolds? Matsumoto conjectured that this happens if and only if $q\ge 3p$; since $\operatorname{rank}(2pE_8\oplus qH)=16p+2q$ and $\operatorname{sign}=16p$, this is equivalent to the inequality $b_2(M)\ge\frac{11}{8}|\operatorname{sign}(M)|$ for every closed smooth spin 4-manifold (no simple-connectivity assumption; the sufficiency direction is realized by $\#_p K3\#_{q-3p}(S^2\times S^2)$, the K3 surface having form $2E_8\oplus 3H$). The conjecture is the last missing piece of the simply connected smooth geography problem after Freedman and Donaldson. Progress has come from gauge theory. Donaldson proved the $p=1$ case (every closed smooth spin 4-manifold with $|\sigma|=16$ satisfies $b_2\ge 22$, with equality realized by the K3 surface) under a 2-torsion hypothesis on $H_1$ later removed by Kronheimer's use of the Pin(2)-symmetry of the Seiberg–Witten equations, the nonlinear elliptic equations that revolutionized 4-manifold topology in 1994. Here Pin(2) is the subgroup $\{e^{i\theta}\}\cup\{je^{i\theta}\}$ of the unit quaternions. Furuta's finite-dimensional approximation turns the Seiberg–Witten map into a Pin(2)-equivariant map between representation spheres, whose stable class — the Bauer–Furuta invariant — depends only on the smooth structure. Furuta used it to prove the 10/8 theorem: $2pE_8\oplus qH$ is spin-realizable only if $q\ge 2p+1$. The homotopy-theoretic core of his argument is the Furuta–Mahowald class of level $(p,q)$: a stable Pin(2)-equivariant map $\gamma:S^{p\mathbb{H}}\to S^{q\widetilde{\mathbb{R}}}$ ($\mathbb{H}$ the 4-dimensional quaternionic Pin(2)-representation, $\widetilde{\mathbb{R}}$ the sign representation) compatible with the equivariant Euler-class inclusions of the fixed points. If such classes existed exactly when $q\ge 3p$, the 11/8-conjecture would follow; Jones's counterexample at $p=5$ already blocked that hope, and Furuta's question — for which $(p,q)$ do Furuta–Mahowald classes exist? — was answered completely by Hopkins–Lin–Shi–Xu: they exist if and only if $q\ge 2p+2$, $2p+3$, or $2p+4$ according to $p\bmod 8$. Their geometric corollary (the 10/8+4 theorem) is that $2pE_8\oplus qH$ is spin-realizable only if $q\ge 2p+2$ for $p\ge 2$, with a mod-8 refinement, and the 'if' part of their theorem proves this is the exact limit of the method: treating the Seiberg–Witten map merely as a continuous Pin(2)-equivariant map can never yield more than $\frac{10}{8}+4$. The 11/8-conjecture itself remains open.

Problem Statement

Determine the truth of Matsumoto's 11/8-conjecture: prove that every closed smooth spin 4-manifold $M$ satisfies $b_{2}(M)\;\ge\;\tfrac{11}{8}\,\bigl|\operatorname{sign}(M)\bigr|,$ or exhibit a counterexample. Equivalently, prove that for every $p\ge 1$ the even indefinite form $2p\,E_{8}\oplus q\,H$ is realizable as the intersection form of a closed smooth spin 4-manifold only if $q\ge 3p$ (the converse being realized by $\#_{p}K3\#_{q-3p}(S^{2}\times S^{2})$, so the open content is entirely the 'only if' direction). The conjecture is stated for manifolds that need not be simply connected; a resolution must hold for, or exhibit a counterexample with, arbitrary fundamental group. Answers that restrict the class of manifolds (for example to simply connected manifolds, definite forms, manifolds with boundary, or spin cobordisms between homology 3-spheres) or that only improve the constant below $\frac{11}{8}$ do not resolve the stated problem. By the Hopkins–Lin–Shi–Xu limit theorem, any proof must use information about the Seiberg–Witten map beyond its Pin(2)-equivariant homotopy class.

References

1. Yukio Matsumoto, 'On the bounding genus of homology 3-spheres', J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 287–318. Zbl 0506.57006. https://zbmath.org/3798672
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8. Jianfeng Lin, 'Pin(2)-equivariant KO-theory and intersection forms of spin 4-manifolds', Algebraic & Geometric Topology 15 (2015) 863–902. DOI: 10.2140/agt.2015.15.863. arXiv:1401.3264. https://doi.org/10.2140/agt.2015.15.863
9. Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu, 'Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant', Communications of the American Mathematical Society 2 (2022) 22–132. DOI: 10.1090/cams/4. arXiv:1812.04052. https://doi.org/10.1090/cams/4
10. William Balderrama, Yueshi Hou, and Shangjie Zhang, '$C_p^n$-equivariant Mahowald invariants', Proceedings of the London Mathematical Society 133 (2026). DOI: 10.1112/plms.70201. https://doi.org/10.1112/plms.70201