Kirby Problem 4.51

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 smooth four-manifold with $\pi_1(X)\cong\mathbb Z$, choose a generator $t$, and put $\Lambda=\mathbb Z[t,t^{-1}]$. Its universal cover $\widetilde X$ is oriented. Let $w(t)\in\{1,-1\}$ record whether the deck transformation preserves that orientation; the group-ring involution is \[ \overline{t}=w(t)t^{-1}. \] Via the Hurewicz identification $\pi_2(X)\cong H_2(\widetilde X;\mathbb Z)$, deck transformations make this a $\Lambda$-module. For compact representatives of classes $a,b$, the equivariant intersection form is \[ \lambda_X(a,b)=\sum_{j\in\mathbb Z}w(t)^j\,(a\cdot t^jb)\,t^j\in\Lambda. \] Only finitely many intersection numbers are nonzero. This is Hermitian: it is linear in the first argument, conjugate-linear in the second, and $\lambda_X(b,a)=\overline{\lambda_X(a,b)}$. For orientable $X$, $w(t)=1$ and the involution is the usual $t\mapsto t^{-1}$.

A form is extended from $\mathbb Z$ if it has a $\Lambda$-basis with integer Gram-matrix entries. Equivalently it is obtained by extending scalars from a symmetric integral form on a finite-rank free abelian group, using the displayed involution.

Statement recorded in the supplied source.

For every closed smooth four-manifold $X$ with $\pi_1(X)\cong\mathbb Z$, is $\lambda_X$ extended from $\mathbb Z$? Precisely, must there be a $\Lambda$-basis $e_1,\ldots,e_r$ of $\pi_2(X)$ such that \[ \lambda_X(e_i,e_j)\in\mathbb Z\qquad(1\le i,j\le r)? \] The target is an integral matrix for the full equivariant form, not only for its augmentation at $t=1$. [S2][S3]

Short English statement

Must the full infinite-cyclic equivariant intersection form of every closed smooth four-manifold have an integral Gram matrix after changing its group-ring basis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2927 (KP-4.51). 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 4.51 and Remarks, PDF page 230. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Stefan Friedl, Ian Hambleton, Paul Melvin and Peter Teichner, Non-smoothable four-manifolds with infinite cyclic fundamental group, arXiv:math/0611077v2 (2007), Introduction, Conjecture 1.3, page 2. https://arxiv.org/pdf/math/0611077v2.