Zeeman collapsibility conjecture

A filled triangle can be dismantled without tearing: remove one edge together with the triangle, leaving two edges meeting at a vertex, then remove each outer vertex together with its edge. Only the meeting vertex remains. Can every finite two-dimensional shape that continuously shrinks to a point be dismantled this way after it is thickened by one interval? Continuous shrinking is called contractibility. Dismantling uses elementary collapses: removing a cell together with a free face that belongs only to that cell. The conjecture permits choosing a suitable finite cell structure on the thickened shape; it does not require every original subdivision to work.[1] [2]

A shape may have a continuous contraction but no available free face from which to start dismantling its given cells. Does adding one product direction always permit a new cell structure with a complete dismantling sequence? For the filled triangle $ABC$, the pairs $(AB,ABC)$, $(A,AC)$, $(B,BC)$ leave the vertex $C$; each first cell is free at its removal. For every compact contractible two-polyhedron $K$, the target is a finite collapsible cell structure on $K\times I$, where $I=[0,1]$. It is an existence statement over cell structures, not a requirement that a specified triangulation collapse. Contractibility allows a homotopy from the identity to a constant map; an elementary collapse removes a cell and its free face.[1] [2]

The triangle already has the required dismantling sequence. Other contractible shapes have no such sequence in their given structure, so continuous shrinking alone is not enough. A known theorem obtains collapsibility after six interval factors for a compact contractible two-dimensional polyhedron. The August 2026 preprint claims the one-interval result for those that embed in a three-manifold and reduces the remaining problem to a specified class. Six factors and the embedding condition do not establish the one-factor conclusion for every shape.[1] [2]

Cohen’s stabilized result supplies collapsibility for $K\times I^6$ in dimension two. The August preprint claims that $K\times I$ is collapsible if $K$ embeds in a three-manifold, and that the universal conjecture reduces to its standard-polyhedron subclass. Those results retain either more factors or an additional hypothesis. They do not establish the one-interval conclusion for an arbitrary compact contractible two-polyhedron. Higher-dimensional noncollapsible products do not constitute counterexamples within this two-dimensional scope.[1] [2]

A continuous shrinking describes a whole motion, whereas a collapse gives a finite certificate made of individually checkable removals. Relating them would say when global deformation information has this stronger combinatorial certificate after a small stabilization. The question also connects to simplifying presentations of the trivial group, in the version that allows adding and removing a matched generator–relation pair. Thus it links the topology of spaces with transformations of algebraic descriptions, rather than only asking whether a dismantling procedure is difficult.[1] [2]

Collapses retain enough combinatorial information to provide a finite sequence of local deformation equivalences, a stronger witness than simply knowing a space is contractible. The interval stabilization asks whether that extra witness can always be obtained with one added direction. Balanced presentations of the trivial group furnish associated two-dimensional complexes, and the stabilized Andrews–Curtis relationship ties algebraic generator–relation moves to this geometric problem. Stabilization in that algebraic statement is essential: it includes adding a generator with a matching relation, not only transformations of a fixed presentation.[1] [2]