On the relation between the Andrews–Curtis and Zeeman collapsibility conjectures

Zeeman's conjecture states that any compact contractible two-polyhedron is $1$-collapsible. In 1987, S. Matveev proved that it holds true for standard non-thickenable polyhedra, provided that the `stable’ version of the Andrews–Curtis conjecture holds. In this talk, I explain an alternative approach that allows one to prove a stronger result. Namely, if a balanced group presentation confirms the original A–C conjecture, the proposed theorem is more specific about how the product of the respective polyhedron with an interval is collapsed.