Kirby Problem 4.15

Kirby Problem 4.15

Definitions and mathematical statement

For a smooth closed oriented four-manifold $M$, the intersection pairing on $H^2(M,{\mathbb{R}})$ is $Q_M(a,b)=\langle a\smile b,[M]\rangle$. Let $b_2$ be its dimension and $\sigma(M)=b_2^+-b_2^-$ its signature. A spin structure exists precisely when the second Stiefel--Whitney class $w_2(TM)$ vanishes. The conjecture is \[ b_2(M)\ge\frac{11}{8}|\sigma(M)| \] for every such spin manifold. No simple-connectedness assumption appears in the selected target. Smoothness is essential: the assertion is not a purely algebraic restriction on arbitrary unimodular bilinear forms or an unrestricted claim for topological four-manifolds.

Formulation and concept references: [S1].

Short English statement

Must the second Betti number of every smooth closed spin four-manifold be at least eleven-eighths the absolute value of its signature?

Research attempt: closure operations and the intersection-form deficit

Status. We prove preservation results and compute the exact numerical deficit on a standard family of forms. These do not prove the conjecture for every smooth spin four-manifold. The smoothness hypothesis is retained throughout. The primary survey [S1] supplies the external facts about the K3 surface and Furuta's theorem used below.

1. Connected sums preserve the desired inequality

For closed oriented connected four-manifolds, Mayer--Vietoris for removal and gluing of balls identifies the real intersection pairing of $M\#N$ with the orthogonal sum of those of $M$ and $N$. Therefore \[ b_2(M\#N)=b_2(M)+b_2(N),\qquad \sigma(M\#N)=\sigma(M)+\sigma(N).\] Spin structures on the punctured manifolds extend across the neck: the sphere $S^3$ has a unique spin structure, so the restrictions can be identified. If both summands obey the target inequality, the triangle inequality gives \[b_2(M\#N)\ge\tfrac{11}{8}(|\sigma(M)|+|\sigma(N)|)\ge\tfrac{11}{8}|\sigma(M\#N)|.\] Orientation reversal changes the sign of signature and leaves $b_2$ unchanged, so it preserves the assertion as well. This argument also shows why cancellation of signatures makes the desired bound easier, not harder.

2. Concrete spin families

Every closed oriented surface $\Sigma_g$ is spin because $w_2(T\Sigma_g)$ is its Euler class modulo two and $2-2g$ is even. The product is spin by the Whitney formula. Künneth gives \[b_2(\Sigma_g\times\Sigma_h)=2+4gh.\] An orientation-reversing diffeomorphism of the first surface, times the identity of the second, reverses the product orientation. Naturality of the intersection pairing then gives $\sigma=-\sigma$, hence signature zero. Thus every such product satisfies the inequality without requiring a simply connected hypothesis.

As an external, standard geometric example, a K3 surface is spin with $b_2=22$ and $\sigma=-16$; it realizes equality. Its orientation reverse does too. Connected sums of $r$ copies with the same orientation satisfy $b_2=22r$ and $|\sigma|=16r$, again equality. Adding $S^2\times S^2$ contributes two to $b_2$ and zero to signature. No statement that every spin four-manifold decomposes into these examples is used.

3. The numerical gap in even indefinite forms

Consider a form consisting of $2a$ copies of a definite $E_8$ form, all of the same sign, and $b$ hyperbolic planes $H$, where $a,b\ge0$. Its rank and absolute signature are \[r=16a+2b,\qquad |s|=16a.\] The target inequality is exactly $b\ge3a$. On a smooth spin manifold with indefinite form, the external $10/8$ theorem quoted in [S1] instead gives $r\ge(10/8)|s|+2$, hence $b\ge2a+1$. For $a=1$ these coincide; for $a\ge2$ the interval $2a+1\le b<3a$ is not eliminated by that estimate alone. This is an exact accounting of the missing strength, rather than an interpolation between the two coefficients.

The algebraic form $2E_8$ has rank 16 and signature 16 and violates $11/8$. It is even and unimodular. This is only an algebraic warning: we have not produced a smooth spin manifold realizing it. In particular, a construction of an intersection matrix without a smooth realization cannot refute the question.

4. Adversarial audit and remaining gap

The connected-sum argument uses absolute values in the correct direction; a lower bound on a sum of absolute signatures would not follow from the absolute signature of the sum. The use of the $10/8$ theorem above is restricted to the indefinite case in the cited formulation. The product calculation and closure results establish genuine families, but no surgery, decomposition, or gauge-theoretic argument here upgrades $2a+1$ to $3a$ for general smooth spin forms. That upgrade, together with the general geometric cases not encoded by these examples, remains unresolved.

Sources

• S1 Ciprian Manolescu, Four-dimensional topology, Section 4.4 $2024$. https://web.stanford.edu/~cm5/4D.pdf. Formulation source.