Kirby Problem 5.12

Definitions and mathematical statement

Let $K$ be a finite $2$-dimensional polyhedron, given by a finite simplicial complex, and suppose $K$ is contractible: its identity map is homotopic to a constant map. Write $I=[0,1]$ and give the product $K\times I$ its natural polyhedral structure, which admits finite triangulations of dimension at most three.

An elementary simplicial collapse removes a simplex $\sigma$ together with a codimension-one face $\tau$ which is a free face: $\sigma$ is the only maximal simplex containing $\tau$. The remaining simplices form a subcomplex. A complex is collapsible if a finite sequence of elementary collapses reduces it to one vertex. For a polyhedron, the assertion uses a compatible finite triangulation, allowing subdivisions.

Does $K\times I$ always collapse to a point in this sense? This is the Zeeman conjecture. The product with an interval adds one dimension to the available collapse process; no further product factor or additional cell attachment is permitted in the requested conclusion.

Each elementary collapse is a particularly controlled deformation retraction, so collapsibility implies contractibility. The hypothesis only supplies contractibility of $K$, and that weaker property alone does not specify a collapse sequence. The conclusion asks for a combinatorial simplification of the product itself. The question ranges over all finite contractible $2$-complexes, without assuming they embed in a $3$-manifold or arise as a spine of one.

Short English statement

Does the product of every finite contractible two-dimensional polyhedron with an interval admit a collapse to a point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3019 (KP-5.12), 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 5.12. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.