Zeeman’s collapsibility conjecture

Prove that for every contractible compact two-dimensional polyhedron K, the product K × [0,1] is collapsible.

Background

Zeeman’s collapsibility conjecture (ZCC) is the principal unresolved conjecture discussed in the paper. It concerns whether taking the product of an arbitrary contractible compact two-dimensional polyhedron with an interval always produces a collapsible polyhedron.

The paper proves that restricting attention to standard polyhedra is sufficient: if ZCC holds for standard polyhedra, then it holds in general. This reduction does not resolve ZCC itself.

References

In 1964, E. C. Zeeman proposed the following conjecture. Let $K$ be a contractible compact two-dimensional polyhedron. Then $K\times\left[0;1\right]$ is collapsible.

— On Zeeman's collapsibility conjecture for non-standard polyhedra  (2608.23331 - Dynnikov, 24 Aug 2026) in Conjecture 1, Section 1 (Introduction)

While ZCC remains unsettled, one can ask a more general question about the smallest~$r\in\mathbb N$ such that any contractible compact two-polyhedron~$K$ is~$r$-collapsible (meaning that~$K\times\left[0;1\right]r$ is collapsible).

— On Zeeman's collapsibility conjecture for non-standard polyhedra  (2608.23331 - Dynnikov, 24 Aug 2026) in Section 1 (Introduction)