Kirby Problem 4.62

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $X$ be a closed connected oriented smooth four-manifold with $b_2^+(X)>1$, and choose a homology orientation to fix invariant signs. Its intersection form on $H^2(X;\mathbb R)$ has positive index $b_2^+$. Write $\chi(X)=\sum_j(-1)^j\dim H_j(X;\mathbb Q)$ for its Euler characteristic and $\sigma(X)=b_2^+-b_2^-$ for its signature.

Let $x\in H_0(X;\mathbb Q)$ be the class of a point and let $D_X$ denote the $SU(2)$ Donaldson polynomial invariant, obtained by evaluating products of the classes $\mu(a)$ on compactified moduli spaces of anti-self-dual connections on principal $SU(2)$ bundles. The polynomial inputs lie in the symmetric algebra on $H_0(X;\mathbb Q)\oplus H_2(X;\mathbb Q)$; $\mu(x)$ has cohomological degree four and $\mu(a)$ for $a\in H_2$ has degree two. Donaldson simple type means \[ D_X(x^2z)=4D_X(z)\qquad\text{for every polynomial input }z. \] This is the two-point-insertion condition from the original definition. In the bundle-index notation, increasing $c_2$ by one adds eight to the moduli dimension, so two degree-four point insertions are required. [S3]

A spin$^c$ structure $\mathfrak s$ has a determinant line bundle with first Chern class $c_1(\mathfrak s)\in H^2(X;\mathbb Z)$. It is a Seiberg--Witten basic structure if the ordinary integer-valued invariant $SW_X(\mathfrak s)$ is nonzero. This scalar invariant is the signed count in dimension zero and, in nonnegative even dimension $d$, the evaluation of the $d/2$ power of the degree-two point class on the monopole moduli space; it is zero by convention in negative or odd expected dimension. [S4] Seiberg--Witten simple type means that every basic structure has expected dimension \[ d_X(\mathfrak s)=\frac{c_1(\mathfrak s)^2-2\chi(X)-3\sigma(X)}4=0. \] Here the square is evaluated on $[X]$.

Statement recorded in the supplied source.

For every $X$ as above, answer both questions: (a) does $X$ have Donaldson simple type, and (b) does $X$ have Seiberg--Witten simple type? Simple connectivity, vanishing first Betti number, and the existence of a symplectic structure are not assumed. [S2]

Short English statement

Must every closed oriented smooth four-manifold with positive intersection index greater than one have both Donaldson and Seiberg–Witten simple type?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2938 (KP-4.62). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.62 and Remarks, PDF page 240. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Ronald Fintushel and Ronald J. Stern, The Blowup Formula for Donaldson Invariants, Annals of Mathematics 143 (1996), 529--546; arXiv:alg-geom/9405002v2, Section 1, pages 1--3, and Section 5. https://arxiv.org/pdf/alg-geom/9405002v2.
• S4 Tsuyoshi Kato, Nobuhiro Nakamura and Kouichi Yasui, The simple type conjecture for mod 2 Seiberg–Witten invariants, arXiv:2009.06791v2 (2021), Introduction, Conjecture 1.1 and its following definition. https://arxiv.org/pdf/2009.06791v2.