Kirby Problem 3.72

Definitions and mathematical statement

Let $\Theta^3_R$ denote the smooth oriented homology cobordism group of $R$-homology three-spheres, for $R=\mathbb Z$ or $\mathbb Q$. A cobordism identifies two classes when its boundary inclusions induce isomorphisms on homology with those coefficients. The group operation is connected sum and the inverse reverses orientation. Changing coefficients gives \[ \Phi:\Theta^3_{\mathbb Z}\longrightarrow\Theta^3_{\mathbb Q}, \qquad K=\ker\Phi. \] Elements of $K$ are represented by integral homology spheres bounding smooth rational homology four-balls.

(a) Does $K$ contain a subgroup isomorphic to $\bigoplus_{j=1}^{\infty}\mathbb Z$? Concretely, seek spheres $Y_j$ representing elements of $K$ such that every finite relation $\sum_j a_j[Y_j]=0$ in integral homology cobordism has all integer coefficients $a_j=0$.

(b) Can such a subgroup $A$ be chosen as a direct summand of $K$? This asks for a decomposition $K\cong A\oplus B$, equivalently a retraction $K\to A$ restricting to the identity on $A$. A splitting in the entire group $\Theta^3_{\mathbb Z}$ would be a stronger conclusion and should be specified separately.

Here $\mathbb Z^\infty$ means the direct sum, whose elements have finite support. An infinite collection of distinct spheres, or infinitely many multiples of a single infinite-order class, does not establish the required independent subgroup. The stronger splitting assertion also requires more than proving independence.

Short English statement

Does the integral-to-rational cobordism kernel contain an infinite free abelian subgroup, and can it split off?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2870 (KP-3.72), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.72. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.