Kirby Problem 3.71

Definitions and mathematical statement

For $R=\mathbb Z$ or $\mathbb F_2$, an $R$-homology sphere is a closed connected oriented smooth three-manifold with the homology of $S^3$ with coefficients in $R$. An $R$-homology cobordism is a compact oriented smooth four-manifold between two such spheres for which both boundary inclusions induce homology isomorphisms with those coefficients. Connected sum gives the group $\Theta^3_R$ of cobordism classes.

Changing coefficients defines a homomorphism \[ \iota:\Theta^3_{\mathbb Z}\longrightarrow\Theta^3_{\mathbb F_2}. \] Is $\ker\iota$ nonzero? Explicitly, is there an integral homology sphere $Y$ which bounds a smooth mod-two homology ball but bounds no smooth integral homology ball? An $R$-homology ball is a compact connected oriented smooth four-manifold $W$ with boundary $Y$ and $H_*(W;R)\cong H_*(B^4;R)$.

Bounding such a ball is equivalent to representing zero in $\Theta^3_R$: remove a small interior ball to obtain a cobordism to $S^3$, or cap that boundary component of a cobordism.

Thus the two filling assertions concern potentially different four-manifolds. The task is to produce a sphere satisfying both, or to prove injectivity of $\iota$. A rational homology ball alone supplies only vanishing with rational coefficients and does not establish mod-two acyclicity. The smooth category and the use of integral homology for the nonvanishing assertion are essential.

Short English statement

Can an integral homology sphere bound a smooth mod-two homology ball but no smooth integral homology ball?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2869 (KP-3.71), 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 3.71. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.