Kirby Problem 4.21

Definitions and mathematical statement

Let $M$ be a closed connected topological four-manifold. A codimension-zero submanifold with boundary is understood to be locally flat along its boundary, so that it has a collar on either side inside $M$. A compact manifold $Y$ is smoothable if its given topology admits a smooth atlas, including compatible boundary charts.

Is there always a decomposition \[ M=Y\cup_\Sigma Z,\qquad Y\cap Z=\partial Y=\partial Z=\Sigma, \] where $Y,Z$ are compact connected topological four-manifolds, $Y$ is smoothable, $Z$ is integrally acyclic, and $\Sigma$ is an integral homology three-sphere? Integrally acyclic means $H_0(Z;\mathbb Z)\cong\mathbb Z$ and $H_i(Z;\mathbb Z)=0$ for every $i>0$; the boundary condition means $H_*(\Sigma;\mathbb Z)\cong H_*(S^3;\mathbb Z)$.

Neither a smoothing nor simple connectivity of $Z$ is required. In particular, acyclicity must not be replaced by contractibility, which would impose a stronger restriction on the discarded piece. The separating homology sphere and the smoothing of $Y$ may both depend on $M$.

The proposed conclusion localizes the obstruction to smoothing a closed manifold inside one acyclic submanifold, without asking that the smooth structure extend across the gluing. The global hypothesis that $M$ is closed is part of the question; a relative version for manifolds with a prescribed outer boundary is not automatically included. No assumption on the fundamental group of $M$ is made.

Short English statement

Can every closed topological four-manifold be split into a smoothable piece and an acyclic piece along a homology sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2897 (KP-4.21), 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 4.21. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.