Kirby Problem 3.69

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.

An integral homology $3$-sphere is a closed oriented smooth $3$-manifold $Y$ with $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$. Two such spheres $Y_0,Y_1$ are integrally homology cobordant if there is a compact connected oriented smooth $4$-manifold $W$ with $\partial W=-Y_0\sqcup Y_1$ for which both inclusions induce isomorphisms on integral homology. Their classes form the abelian group $\Theta^3_{\mathbb Z}$ under connected sum, with zero $[S^3]$ and inverse $[-Y]$. The Rokhlin homomorphism $\mu:\Theta^3_{\mathbb Z}\to\mathbb Z/2\mathbb Z$ is $\mu([Y])=\sigma(X)/8\pmod2$, where $X$ is any smooth compact spin $4$-manifold bounded by $Y$ and $\sigma$ is its intersection-form signature. A nontrivial torsion element means $x\ne0$ with $mx=0$ for some $m\ge2$. For $n\ge1$, $nz$ means the sum of $n$ copies of $z$.

Statement recorded in the supplied source.

Resolve all four questions:

1. Determine the isomorphism type of $\Theta^3_{\mathbb Z}$.
2. Does $\Theta^3_{\mathbb Z}$ contain a nontrivial torsion element?
3. Does it contain a torsion element $x$ with $\mu(x)=1$?
4. Does there exist $x=[Y]$ such that the set \[ \{n\in\mathbb N:\ \exists z_n\in\Theta^3_{\mathbb Z},\ x=nz_n\} \] is infinite?

Part (d) is the literal infinitely-many-$n$ condition in the source. It permits $x=0$, which satisfies the condition by taking $z_n=0$ for every $n$. Requiring a nonzero class, requiring divisibility by every $n$, or passing to the quotient by torsion are separate stronger formulations. [S2]

Short English statement

Determine the integral homology-cobordism group, whether it has nontrivial torsion or torsion with Rokhlin invariant one, and its stated infinitely-many-divisors condition. The literal last condition is already satisfied by the zero class.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2867 (KP-3.69). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.69, p. 181 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.