The 11/8-conjecture (Matsumoto): $b_2\ge\frac{11}{8}|\sigma|$ for closed smooth spin 4-manifolds

The 11/8-conjecture of Matsumoto asserts that every closed smooth spin 4-manifold $X$ satisfies $b_2(X)\ge\frac{11}{8}|\sigma(X)|$, where $b_2$ is the second Betti number and $\sigma$ the signature; equivalently, the even indefinite unimodular lattice $2pE_8\oplus qH$ occurs as the intersection form of such a manifold only when $q\ge 3p$, the borderline being realized by connected sums of the K3 surface. It is the last missing piece of the geography problem for simply connected smooth 4-manifolds. The best general bound remains Furuta's 10/8 theorem ($b_2\ge\frac{10}{8}|\sigma|+2$), refined residue-class-wise by Furuta-Kametani and by Hopkins-Lin-Shi-Xu, whose "10/8+4" theorem also proves that the stable-homotopy (Furuta-Mahowald) route is capped at $\frac{10}{8}|\sigma|+4$. The conjecture is equivalent to a lower bound on the bounding genus of homology 3-spheres with vanishing Rokhlin invariant.

Background

For a closed, oriented, smooth 4-manifold $X$, the cup product makes $H^2(X;\mathbb{Z})$ modulo torsion into a unimodular (determinant $\pm 1$) symmetric bilinear lattice $Q_X$, called the intersection form; its rank is the second Betti number $b_2(X)$ and its signature (number of positive minus negative eigenvalues) is $\sigma(X)$. The geography problem asks which unimodular forms arise as $Q_X$ for closed simply connected smooth 4-manifolds. Freedman's topological classification shows that in the topological category an even form is realized exactly when the Kirby-Siebenmann invariant (the obstruction to a topological 4-manifold admitting any smooth structure) equals $\sigma/8$ modulo 2, by a unique simply connected manifold; in the smooth category, Donaldson's diagonalizability theorem forces every definite form realized by a closed simply connected smooth 4-manifold to be $\pm I$. By the classical Hasse-Minkowski classification of indefinite unimodular forms, the remaining indefinite forms are the odd ones (diagonal $\pm 1$, all smoothly realized by connected sums of $\mathbb{C}P^2$ and $\overline{\mathbb{C}P}^2$) and the even ones (a unimodular form is even when $Q(x,x)$ is even for every $x$, and odd otherwise), which are exactly $kE_8\oplus lH$: here $E_8$ is the positive-definite even unimodular lattice of rank 8 and signature 8, and $H$ is the hyperbolic plane $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ of rank 2 and signature 0.

An oriented 4-manifold is spin when its second Stiefel-Whitney class $w_2\in H^2(X;\mathbb{Z}/2)$ (the characteristic class obstructing a spin structure) vanishes; by Wu's formula this is equivalent to $Q_X$ being even, and Rohlin's theorem forces $\sigma(X)\equiv 0\pmod{16}$ for any closed smooth spin 4-manifold. So the intersection form of a closed smooth spin 4-manifold is $2pE_8\oplus qH$ up to sign, with $p\ge 0$. The realization direction is easy: the K3 surface, a simply connected complex surface with intersection form $2E_8\oplus 3H$, $b_2=22$ and $\sigma=-16$, shows that the connected sum $\#_p K3\#(q-3p)(S^2\times S^2)$, i.e. $p$ copies of the K3 surface together with $q-3p$ copies of $S^2\times S^2$, realizes $2pE_8\oplus qH$ whenever $q\ge 3p$.

Matsumoto's 11/8-conjecture, posed in 1982 in the language of the bounding genus of homology 3-spheres, asserts that this construction is optimal: the form $2pE_8\oplus qH$ with $p\ge 1$ is realized by a closed smooth spin 4-manifold only if $q\ge 3p$. Since $b_2=16p+2q$ and $|\sigma|=16p$ for such a form, this is equivalent to the inequality $b_2(X)\ge\frac{11}{8}|\sigma(X)|$ for every closed smooth spin 4-manifold $X$ (no simple connectivity is needed), with the K3 surface attaining equality. Together with Freedman's and Donaldson's theorems, the conjecture is the last missing piece of the simply connected smooth geography problem: it would determine exactly which even indefinite forms are smoothly realizable.

The known partial results come from gauge theory. Donaldson proved the case $p=1$ (a single $E_8$ pair) under the assumption that $H_1(X;\mathbb{Z})$ has no 2-torsion, and that assumption was removed by Kronheimer (unpublished 1994 lecture, as documented by Hopkins-Lin-Shi-Xu) using Pin(2)-symmetries of the Seiberg-Witten equations; here Pin(2), the normalizer of the circle subgroup in the unit quaternions, is an extra symmetry group of the monopole equations on a spin 4-manifold. Furuta's 10/8 theorem, proved by finite-dimensional approximation of the monopole equations (replacing the Seiberg-Witten map by a Pin(2)-equivariant map between finite-dimensional representation spheres, the Bauer-Furuta invariant), gives $b_2\ge\frac{10}{8}|\sigma|+2$, equivalently $q\ge 2p+1$. Furuta and Kametani obtained the first residue-class refinements. Hopkins, Lin, Shi and Xu then completely solved Furuta's problem on Pin(2)-equivariant stable maps (via the Pin(2)-equivariant Mahowald invariant) and proved the full refinement: for $p\ge 2$, $2pE_8\oplus qH$ is spin-realizable only if $q\ge 2p+2$ when $p\equiv 1,2,5,6\pmod 8$, $q\ge 2p+3$ when $p\equiv 3,4,7\pmod 8$, and $q\ge 2p+4$ when $p\equiv 0\pmod 8$; in particular every closed simply connected smooth spin 4-manifold other than $S^4$, $S^2\times S^2$ and the K3 surface satisfies $b_2\ge\frac{10}{8}|\sigma|+4$. Crucially, the same paper proves that this is the limit of the method: using only the existence of the relevant Furuta-Mahowald classes, no bound stronger than $\frac{10}{8}|\sigma|+4$ can be obtained, so a proof of the 11/8-conjecture requires finer geometric or analytic properties of the Seiberg-Witten map beyond its equivariant stable homotopy class; Furuta, Kametani, Matsue and Minami had earlier shown that nilpotency phenomena in the Bauer-Furuta invariants obstruct any proof of 11/8 from that theory alone.

There is an equivalent formulation on the 3-dimensional side. An integral homology 3-sphere is a closed 3-manifold $Y$ with the same integral homology as $S^3$; its Rokhlin invariant $\mu(Y)\in\mathbb{Z}/2$ is $\sigma(W)/8$ modulo 2 for any compact spin 4-manifold $W$ with boundary $Y$. Matsumoto's bounding genus $|Y|$ is the minimal $n$ such that $Y$ bounds a homologically 1-connected (i.e. $H_1=0$) spin 4-manifold with intersection form $nH$ (finite exactly when $\mu(Y)=0$); Tange's invariant $\mathfrak{ds}(Y)$ is the maximum of $b_2(W)/8$ over definite spin boundings $W$ of $Y$. Tange proved that Matsumoto's conjecture is equivalent to the statement $|Y|\ge\frac{3}{2}\mathfrak{ds}(Y)$ for every homology 3-sphere with $\mu(Y)=0$ and $\mathfrak{ds}(Y)<\infty$. Bauer proposed attacking the simply connected case by cutting a hypothetical counterexample along homology 3-spheres; Manolescu's $\kappa$-invariant of homology 3-spheres, constructed from Pin(2)-equivariant Seiberg-Witten Floer K-theory, and his Floer $K_G$-split condition (with $G=\mathrm{Pin}(2)$; a splitting property of the equivariant Seiberg-Witten Floer $K$-theory of a homology 3-sphere, defined in his paper) give the sharpest obstruction in this direction: no counterexample can admit such a decomposition with all cutting homology spheres Floer $K_G$-split. However, many Brieskorn spheres (links of complex surface singularities, such as $\Sigma(2,3,11)$, forming a standard explicit family of homology 3-spheres) are provably not $K_G$-split, so this route also remains open.

Problem Statement

Prove or refute the 11/8-conjecture of Matsumoto in its standard full generality: every closed, smooth, oriented, spin 4-manifold $X$ satisfies

$b_2(X)\;\ge\;\frac{11}{8}\,\left|\sigma(X)\right|.$

Equivalently: for every $p\ge 1$ and $q\ge 0$, the even indefinite unimodular form $2pE_8\oplus qH$ arises as the intersection form of a closed smooth spin 4-manifold if and only if $q\ge 3p$ (the "if" direction being realized by $\#_p K3\#(q-3p)(S^2\times S^2)$). Equivalently, in the homology-3-sphere form (Tange): every integral homology 3-sphere $Y$ (a closed 3-manifold with the same integral homology as $S^3$) with Rokhlin invariant $\mu(Y)=0$ and $\mathfrak{ds}(Y)<\infty$ satisfies $|Y|\ge\frac{3}{2}\mathfrak{ds}(Y)$; here $\mu(Y)=\sigma(W)/8\pmod 2$ for any compact spin 4-manifold $W$ with $\partial W=Y$, $|Y|$ is the bounding genus (the minimal $n$ such that $Y$ bounds a homologically 1-connected, i.e. $H_1=0$, spin 4-manifold with intersection form $nH$), and $\mathfrak{ds}(Y)$ is the maximum of $b_2(W)/8$ over definite spin boundings $W$ of $Y$. A resolution consists of either a proof of the inequality for all closed smooth spin 4-manifolds, or an explicit closed smooth spin 4-manifold $X$ with $b_2(X)<\frac{11}{8}|\sigma(X)|$ (equivalently, with intersection form $2pE_8\oplus qH$ and $q\le 3p-1$). Partial bounds such as Furuta's 10/8 theorem or the residue-refined 10/8+4 theorem do not resolve the problem.

References

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