Kirby Problem 4.17

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.

For a closed simply connected oriented smooth four-manifold $X$, the intersection form is \[ Q_X(a,b)=\langle a\smile b,[X]\rangle,\qquad a,b\in H^2(X;\mathbb R). \] Its positive and negative indices are $b_2^+(X)$ and $b_2^-(X)$: the dimensions of maximal positive- and negative-definite subspaces. A smooth four-manifold is irreducible if every smooth connected-sum decomposition $X\cong X_1\#X_2$ has a summand that is a homotopy four-sphere, that is, a manifold homotopy equivalent to $S^4$. Irreducibility does not require that this homotopy sphere already be known to be diffeomorphic to standard $S^4$. [S3]

Statement recorded in the supplied source.

Does there exist an irreducible closed simply connected oriented smooth four-manifold $X$ for which both $b_2^+(X)$ and $b_2^-(X)$ are even integers? \[ \exists X:\quad b_2^+(X)\equiv b_2^-(X)\equiv0\pmod2. \] No symplectic or complex structure is required. [S2] With this literal formulation the answer is affirmative for standard $S^4$: both indices are zero, and any connected-sum decomposition has summands with zero second homology and trivial fundamental group, hence homotopy four-spheres. A requirement that the even indices be positive would be an additional hypothesis not present in the recorded question.

Short English statement

Is there a closed simply connected smooth irreducible four-manifold whose intersection form has both positive and negative indices even?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2893 (KP-4.17). 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 4.17 and Remarks, PDF page 203. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 M. J. D. Hamilton and D. Kotschick, Minimality and irreducibility of symplectic four-manifolds, IMRN (2006), Article 35032; arXiv:math/0509204v2, Section 1.2, page 2. https://arxiv.org/pdf/math/0509204v2.