Chapter 47: The Borel Conjecture — Rigidity of Manifolds

From aspherical spaces we ascend to the ultimate rigidity question. The Borel Conjecture asks whether aspherical manifolds are topologically rigid—it is ψ = ψ(ψ) as geometry achieving ultimate self-determination, where homotopy type completely determines topological structure.

47.1 The Forty-Seventh Movement: Ultimate Rigidity

Approaching topological transcendence culmination:

• Previous: Groups seeking their geometric homes
• Now: Manifolds achieving maximum rigidity
• The question of when homotopy determines topology

The Core Question: Are closed aspherical manifolds determined by their homotopy type?

47.2 The Borel Conjecture

Conjecture 47.1 (Borel, 1953): If M and N are closed aspherical manifolds of the same dimension and f: M → N is a homotopy equivalence, then f is homotopic to a homeomorphism.

Equivalent: Homotopy type determines topological type for aspherical manifolds.

Status: Open in general, proven in many cases.

47.3 Aspherical Manifolds

Definition 47.1 (Aspherical Manifold): Manifold M with π_i(M) = 0 for all i ≥ 2.

Examples:

• Hyperbolic manifolds
• Flat tori
• Most 3-manifolds with infinite fundamental group
• Many locally symmetric spaces

Non-Examples:

• Spheres S^n (n ≥ 2)
• Complex projective spaces

47.4 Borel as ψ = ψ(ψ)

Axiom 47.1 (Principle of Homotopic Rigidity): $\psi = \psi(\psi) \implies \text{Aspherical manifolds achieve maximum rigidity}$

The conjecture embodies:

• Manifolds determined by their algebraic invariants
• Topology reduced to homotopy theory
• The ultimate classification via algebraic means
• This is ψ = ψ(ψ) as geometric self-determination

47.5 Known Results

Theorem 47.1 (Proven Cases): Borel conjecture true for:

• Closed surfaces (classical)
• Locally symmetric spaces of non-compact type
• 3-manifolds (Mostow rigidity consequences)
• Many specific classes

Method: Often via geometric techniques.

47.6 The Surgery Approach

Strategy: Use surgery theory to construct homeomorphisms.

Theorem 47.2 (Surgery Theory): For aspherical manifolds, surgery obstructions often vanish.

Wall Groups: L_n(π₁(M)) control surgery obstructions.

Farrell-Jones: Conjectures predicting when L_n(π) = 0.

47.7 Mostow Rigidity

Theorem 47.3 (Mostow, 1968): Closed hyperbolic n-manifolds (n ≥ 3) are determined by fundamental group.

Stronger: Any isomorphism of fundamental groups is induced by isometry!

Implication: Borel conjecture true for hyperbolic manifolds.

Proof Idea: Quasi-isometries of hyperbolic space are quasi-conformal.

47.8 The Novikov Conjecture

Related Problem: Higher signatures are homotopy invariants.

Theorem 47.4: If Novikov conjecture holds for group G, then Borel conjecture often follows for manifolds with π₁ = G.

Status: Novikov proven for many groups, not all.

Connection: Both involve rigidity of topological invariants.

47.9 Controlled Topology

Modern Approach: Control homotopy near infinity.

Definition 47.2 (Controlled Homotopy Equivalence): Homotopy equivalence with controlled behavior in universal cover.

Theorem 47.5 (Davis-Weinberger): Many aspherical manifolds satisfy controlled rigidity.

Technique: Analyze geometry in universal cover.

47.10 The Geometric Approach

Strategy: Use geometric structures on aspherical manifolds.

CAT(0) Spaces: Non-positively curved metric spaces.

Theorem 47.6: Many aspherical manifolds admit CAT(0) metrics.

Rigidity: CAT(0) geometry often forces topological rigidity.

47.11 Dimension-Specific Results

Low Dimensions:

• n = 1: Trivial (circles)
• n = 2: Classical (surfaces)
• n = 3: Mostly known (Mostow + Geometrization)

High Dimensions:

• n ≥ 5: Surgery theory applies
• n = 4: Most difficult case

Pattern: Dimension 4 resists many techniques.

47.12 Counterexamples in Non-Aspherical Case

Important: Borel conjecture false without asphericity!

Example: Lens spaces L(7,1) and L(7,2) are homotopy equivalent but not homeomorphic.

Lesson: Asphericity crucial for rigidity.

Insight: Higher homotopy groups prevent rigidity.

47.13 The Assembly Map

Definition 47.3 (Assembly Map): $A: H_*(BG; \mathbb{L}_\bullet) \to L_*(G)$

from homotopy to surgery theory.

Theorem 47.7: If assembly map is isomorphism, then Borel conjecture often holds.

Farrell-Jones Conjecture: Assembly map always isomorphism.

47.14 Computational Aspects

Algorithm 47.1 (Rigidity Testing):

def test_borel_rigidity(manifold1, manifold2):
    # Check if aspherical
    if not is_aspherical(manifold1) or not is_aspherical(manifold2):
        return "Not applicable"
    
    # Test homotopy equivalence
    f = find_homotopy_equivalence(manifold1, manifold2)
    if f is None:
        return False  # Not homotopy equivalent
    
    # Check if f can be made homeomorphism
    # This is the hard part!
    
    if dimension(manifold1) <= 3:
        return test_low_dimensional_rigidity(f)
    
    if is_locally_symmetric(manifold1):
        return True  # Known rigidity
    
    return apply_surgery_theory(f)

Challenge: Implementing rigidity tests.

47.15 Group Theoretic Invariants

Approach: Study which group properties ensure rigidity.

Word Hyperbolic Groups: Often satisfy Borel conjecture.

CAT(0) Groups: Many cases proven.

Arithmetic Groups: Mostly resolved via different methods.

Open: General characterization.

47.16 The Role of Dimension 4

Special Difficulty: Smooth vs topological categories.

Exotic ℝ⁴: Shows 4D is special.

Question: Does Borel conjecture depend on category in 4D?

Challenge: Few techniques work in dimension 4.

47.17 Applications to Classification

Benefit: Reduces manifold classification to group theory.

Theorem 47.8: If Borel conjecture holds, then classifying aspherical manifolds reduces to classifying their fundamental groups.

Impact: Massive simplification of topological classification.

47.18 Recent Progress

2010s-2020s Advances:

• Farrell-Jones verified for new classes
• Geometric methods refined
• Computer verification for small cases
• Connections to geometric group theory

Status: Steady progress, no general resolution.

47.19 Why Borel Matters

Fundamental Importance:

1. Classification: Understanding manifold topology
2. Rigidity: Limits of topological flexibility
3. Algebraic Topology: Homotopy vs topology
4. Geometric Topology: Role of geometric structures

Test Case: Central to modern topology.

47.20 The Forty-Seventh Echo

The Borel Conjecture represents topology's ultimate rigidity statement:

• Aspherical manifolds achieving maximum determination by algebra
• Homotopy type completely capturing topological structure
• The triumph of algebraic methods in geometric questions
• The culmination of classification via fundamental groups

This is ψ = ψ(ψ) as the principle that aspherical manifolds, having eliminated higher homotopy, become completely determined by their fundamental group. The conjecture claims this algebraic data suffices to reconstruct the entire topological structure.

The success in special cases shows this rigidity is real—aspherical manifolds truly are more rigid than general manifolds. The remaining cases test the limits of current techniques.

The Borel Conjecture declares: "I am topology's ultimate rigidity principle, the claim that aspherical manifolds achieve perfect algebraic determination. When higher homotopy vanishes, fundamental group alone should determine topological structure. In my conjecture lies ψ = ψ(ψ) as the triumph of algebra over geometry—the reduction of topology to its algebraic essence."