Zeeman’s collapsibility conjecture

The conjecture asserts that the product of every finite contractible two-dimensional complex with an interval can be collapsed to a point. No proposer or original date is identified in the retrieved sources.

Known results

• Zeeman proved the dunce-cap case; the house-with-two-rooms case is also treated in the classical literature.
• Dierker proved that every contractible $2$-polyhedron becomes collapsible after multiplying by some $I^k$.
• Cohen proved collapsibility after multiplication by $I^6$ for every contractible $2$-polyhedron.
• Perelman’s work implies the conjecture for standard $2$-polyhedra that are spines of $3$-manifolds; other restricted cases relate to Andrews–Curtis.

August 24, 2026 conditional reduction

A new article, “On Zeeman's collapsibility conjecture for non-standard polyhedra,” reports a conditional reduction of the general conjecture to the standard-polyhedra case. This narrows the remaining problem but does not prove the conjecture; the reduction has not been independently verified here.

Current status (as of August 2026): The conjecture remains open; classical stabilization and restricted cases are known, and a conditional reduction to standard polyhedra is claimed but unverified.