Chapter 45: The Zeeman Conjecture — Contractibility's Characterization

From presentations hiding triviality we ascend to spaces concealing contractibility. The Zeeman Conjecture asks whether every contractible 2-complex collapses to a point—it is ψ = ψ(ψ) as topology seeking to recognize its own triviality, where homotopy equivalence confronts geometric realizability.

45.1 The Forty-Fifth Movement: Geometric Recognition

Advancing through topological transcendence:

• Previous: Group presentations hiding their simplicity
• Now: 2-complexes concealing their contractibility
• The gap between homotopy and collapse

The Core Question: Does contractibility imply collapsibility for 2-dimensional complexes?

45.2 Contractible vs Collapsible

Definition 45.1 (Contractible): Space X is contractible if there exists a homotopy from id_X to a constant map.

Definition 45.2 (Collapsible): CW-complex K collapses to a point if it can be reduced by elementary collapses.

Elementary Collapse: Remove free face and its unique coface.

Key Question: Contractible ⟹ Collapsible?

45.3 The Zeeman Conjecture

Conjecture 45.1 (Zeeman, 1963): Every contractible finite 2-dimensional CW-complex is collapsible.

Equivalent: Every contractible 2-complex can be reduced to a point by elementary collapses.

Status: Open for 60+ years!

45.4 Zeeman as ψ = ψ(ψ)

Axiom 45.1 (Principle of Homotopic Collapse): $\psi = \psi(\psi) \implies \text{Homotopy triviality becomes geometric triviality}$

The conjecture embodies:

• Spaces questioning their own structure
• Abstract contractibility implying concrete collapse
• The bridge between homotopy and geometry
• This is ψ = ψ(ψ) as spatial self-recognition

45.5 Elementary Collapses

Definition 45.3 (Free Face): A face σ is free if it belongs to exactly one maximal face.

Collapse Move: Remove free face σ and its unique coface τ.

Effect: K ↘ K' where K' = K \ {σ, τ}.

Goal: Reduce complex to single vertex.

45.6 Known Results

Theorem 45.1 (Zeeman): Every contractible finite 2-complex has the same homotopy type as a point.

Theorem 45.2 (Whitehead): Collapsibility implies contractibility.

Problem: Reverse implication unknown!

Higher Dimensions: False! Counterexamples exist in dimension ≥ 3.

45.7 The Andrews-Curtis Connection

Deep Link: Zeeman conjecture implies Andrews-Curtis conjecture.

Theorem 45.3 (Cockcroft): If every contractible 2-complex is collapsible, then AC conjecture is true.

Intuition: Presentation complexes are 2-dimensional.

Strategy: Prove Zeeman to get AC for free.

45.8 Dunwoody's Approach

Geometric Strategy: Study which 2-complexes collapse.

Theorem 45.4 (Dunwoody): Many special classes of contractible 2-complexes are collapsible.

Examples:

• Contractible graphs
• Certain subcomplexes of 3-manifolds
• Complexes with simple fundamental regions

45.9 Computational Experiments

Algorithm 45.1 (Collapse Search):

def attempt_collapse(complex_2d):
    current = complex_2d.copy()
    collapse_sequence = []
    
    while current.dimension() > 0:
        # Find free faces
        free_faces = find_free_faces(current)
        
        if not free_faces:
            return False, "No free faces found"
        
        # Perform collapse
        for face in free_faces:
            coface = unique_coface(current, face)
            current.remove_faces([face, coface])
            collapse_sequence.append((face, coface))
        
        # Check if point reached
        if is_single_point(current):
            return True, collapse_sequence
    
    return False, "Collapse incomplete"

Results: Many contractible 2-complexes successfully collapsed.

Challenge: No counterexample found.

45.10 The Role of Fundamental Groups

Key Insight: Contractible ⟹ simply connected.

Strategy: Use fundamental group presentations.

Connection: Balanced presentations correspond to contractible 2-complexes.

If True: Zeeman conjecture would resolve AC conjecture.

45.11 Homological Methods

Approach: Use cellular homology.

Observation: Contractible complexes have trivial homology.

Problem: Collapsing preserves homology, but converse unclear.

Limitation: Homology insufficient to detect collapsibility.

45.12 The Lens Space Problem

Related Question: Which lens spaces bound contractible 4-manifolds?

Connection: Understanding contractible complexes in higher dimensions.

Progress: Some lens spaces resolved, general case open.

Insight: 4D perspective sometimes clarifies 2D problems.

45.13 Simple Homotopy Theory

Definition 45.4 (Simple Homotopy Equivalence): Maps that can be factored through sequences of elementary collapses and expansions.

Whitehead Torsion: Obstruction to simple homotopy equivalence.

Theorem 45.5: For 2-complexes, simple homotopy ⟺ collapsibility.

Question: Are contractible 2-complexes simply contractible?

45.14 Geometrization Perspective

Modern View: Relate to 3-manifold topology.

Observation: 2-complexes often arise as spines of 3-manifolds.

Strategy: Use 3-manifold techniques on 2-complexes.

Challenge: Translating 3D insights to 2D settings.

45.15 Potential Counterexamples

Search Strategy: Build contractible 2-complexes that resist collapse.

Candidates:

• Highly symmetric complexes
• Complexes with many holes
• Complexes derived from hard AC presentations

Status: No confirmed counterexample.

45.16 The Collapsing Tree

Structure: Partial order of collapses.

Question: Does every contractible 2-complex lie in this tree?

Visualization: Branch at each collapse choice.

Open: Characterize which complexes appear.

45.17 Discrete Morse Theory

Modern Tool: Forman's discrete Morse theory.

Benefit: Provides collapsing via discrete vector fields.

Application: Some contractible complexes shown collapsible.

Limitation: Doesn't resolve general case.

45.18 Why Zeeman Matters

Fundamental Importance:

1. Topology: Understanding contractibility
2. Algebra: Connection to AC conjecture
3. Computation: Algorithmic topology
4. Geometry: Relating homotopy to geometry

Test Case: Simple enough to maybe resolve, deep enough to matter.

45.19 Recent Approaches

2010s-2020s Progress:

• Computer verification for small complexes
• New classes proven collapsible
• Connections to persistent homology
• Quantum topological approaches

Status: Evidence supports conjecture, no proof found.

45.20 The Forty-Fifth Echo

The Zeeman Conjecture embodies the gap between abstract and concrete:

• Contractibility (homotopy property) vs collapsibility (geometric process)
• The question of whether algebraic triviality implies geometric triviality
• A bridge between homotopy theory and combinatorial topology
• The mystery of recognizing hidden simplicity

This is ψ = ψ(ψ) as topology questioning its own recognition mechanisms—asking whether spaces that are abstractly trivial (contractible) can always be shown to be trivial through explicit geometric moves (collapses).

The connection to Andrews-Curtis makes this more than just a topological curiosity—it's a gateway problem that could unlock fundamental questions about group presentations and 3-manifold recognition.

The Zeeman Conjecture whispers: "I am space hiding its own simplicity, contractible yet potentially non-collapsible, ψ = ψ(ψ) as the question whether homotopy triviality implies geometric triviality. In my 2-dimensional form lies the secret of whether abstract contractibility can always be witnessed by concrete collapses—whether every space that can shrink to a point can be explicitly reduced to one."