PL topology · collapsibility · finite 2-complexes · Andrews–Curtis conjecture

Zeeman Conjecture

Collaboration beta

Does multiplying any finite contractible two-dimensional complex by an interval always make it collapsible? Restricted algebraic and fake-surface routes do not yet bridge back to arbitrary complexes.

K finite contractible 2-polyhedron \Rightarrow K \times I ↘ *

Known results and sources

The Zeeman conjecture asks whether one interval factor always makes a finite contractible 2-complex collapsible after subdivision.

Research problem

Exact mathematical statement

For every finite contractible two-dimensional polyhedron $K$ , does $K\times I$ , where $I=[0,1]$ , admit a triangulation—or equivalently a suitable subdivision in the PL formulation—that collapses to one vertex?

K\times I\searrow *.

A counterexample must obstruct collapsibility after every allowed subdivision; a single noncollapsible triangulation is not enough.

Problem infographic

Problem at a glance

A four-stage topology explainer separates the exact K times I target, restricted fake-surface algebra, unresolved weighted cores, and the missing arbitrary-K cylinder transfer. — Restricted fake-surface and stable Andrews–Curtis reductions do not by themselves produce a collapse of K×I for arbitrary contractible K.

Current mathematical picture

Where work on Zeeman Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTreating stable Andrews–Curtis or 3-deformation as an automatic collapse of arbitrary K×I

A formal Tietze or stable-AC sequence without a common subdivision and free-face order does not witness cylindrical collapsibility. A normalization-specific transfer, relative prism simulation, rank-three weighted-core theorem, exact correlated-defect reduction, or subdivision-stable obstruction remains viable.

Route status · Narrowed route

Main reductionCurrent reduction

Normalize a contractible fake surface to a balanced cubic presentation, use a short square-free relator, convert saturated systems to signed doubling chains and weighted cycle cores, then attack rank-three cores—while separately proving an interval-compatible transfer from arbitrary K.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve an actual one-interval transfer or direct cylinder realization from an arbitrary K to a collapsible K×I.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Zeeman Conjecture in numbers

655retained lines of mathematical investigation655 in the current working snapshot

Argument development: 533 · 81%
Explored or eliminated routes: 27 · 4%
Computational analysis: 4 · 1%
Open obligations: 63 · 10%
Definitions and setup: 28 · 4%

7selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byE. C. ZeemanProposed1964Status checkedAug 20, 2026

Source recordedAug 20, 2026

Recommended next task

Prove an actual one-interval transfer or direct cylinder realization from an arbitrary K to a collapsible K×I.

Suggested move: Trace the standard 3-deformation to a non-spine fake surface one elementary move at a time and record which moves admit compatible subdivisions and explicit free-face orders.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeTreating stable Andrews–Curtis or 3-deformation as an automatic collapse of arbitrary K×I

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove an actual one-interval transfer or direct cylinder realization from an arbitrary K to a collapsible K×I.Suggested move: Trace the standard 3-deformation to a non-spine fake surface one elementary move at a time and record which moves admit compatible subdivisions and explicit free-face orders.

Ready to work on

Eliminate every rank-three weighted triangle without relying only on the exponent matrix.Suggested move: Treat all three label permutations and both block orders, seeking an explicit stable-AC sequence to a two-branch core, defining generator, removable component, or thickenable presentation.

Ready to work on

Control the nonsaturated common-domino defect created by one cubic pivot.Suggested move: Build the exact marked defect graph and prove another pivot, restoration of a smaller cubic system, thickenability, or a bounded-rank reduction.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 20, 2026

Current statusOpen conjecture

The conjecture that K×I is collapsible for every finite contractible two-dimensional complex K remains open. Known collapsible products for named complexes and structural reductions do not establish the uniform statement.

[1][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2021
Authoritative summaryKupers surveyed the conjecture, its basic implications, and representative examples while recording the general statement as open.[2]
1964
Historical sourceZeeman introduced the dunce-hat setting and the product-with-an-interval collapsibility question that became the conjecture.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusZeeman Conjecture

Solved special caseDunce-hat and other explicit contractible 2-complex examples

The dunce hat and other named contractible complexes provide important collapsible-product examples, but finitely many examples do not prove the every-complex statement.

[1][2]

Logical consequenceThree-dimensional Poincaré and simple-homotopy questions

The classical literature records strong implications of the Zeeman conjecture, including its relation to the three-dimensional Poincaré conjecture; those implications do not run backward to provide a proof of Zeeman's statement.

[1][2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA formal model of finite contractible 2-complexes, products with an interval, subdivisions, and elementary collapses.
Formalization targetChecked simple-homotopy infrastructure connecting local collapse moves to the global product complex.
Formalization targetA uniform proof for arbitrary finite contractible 2-complexes; verified named examples are insufficient.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.