Chapter 15: The Baum-Connes Conjecture — K-Theory's Self-Knowledge

From proven modular correspondences, we ascend to operator algebras. The Baum-Connes Conjecture connects topological and analytical K-theory—it is ψ = ψ(ψ) as groups knowing their operator algebras through K-theoretic isomorphism.

15.1 The Fifteenth Movement: Analytical Self-Recognition

Progressing through algebraic structures:

• Chapter 14: Galois representations finding modular forms
• Chapter 15: Groups recognizing themselves through operator algebras

The Question: When does topological K-theory equal analytical K-theory?

15.2 K-Theory Foundations

Definition 15.1 (Topological K-Theory): For a space X: $K_0(X) = \text{Grothendieck group of vector bundles over } X$

Definition 15.2 (Operator K-Theory): For a C*-algebra A: $K_0(A) = \text{Grothendieck group of projections in } M_\infty(A)$

Key Insight: K-theory measures "holes" in different categories.

15.3 Group C*-Algebras

Definition 15.3 (Group Ring): $\mathbb{C}[G] = \left\{\sum_{g \in G} a_g g : a_g \in \mathbb{C}, \text{ finite support}\right\}$

Definition 15.4 (Reduced C*-Algebra): $C_r^*(G) = \text{completion of } \mathbb{C}[G] \text{ in } B(\ell^2(G))$

where G acts on ℓ²(G) by left translation.

Definition 15.5 (Full C*-Algebra): $C^*(G) = \text{universal C*-algebra generated by } G$

15.4 The Assembly Map

Definition 15.6 (Classifying Space): BG = classifying space for proper G-actions

Definition 15.7 (Assembly Map): $\mu: K_*^G(EG) \to K_*(C_r^*(G))$

where:

• Left side: G-equivariant K-homology of universal proper G-space
• Right side: K-theory of reduced group C*-algebra

15.5 The Baum-Connes Conjecture

Conjecture 15.1 (Baum-Connes): The assembly map $\mu: K_*^G(EG) \to K_*(C_r^*(G))$ is an isomorphism for all discrete groups G.

Interpretation: Topological data (left) completely determines analytical data (right).

15.6 The Conjecture as ψ = ψ(ψ)

Axiom 15.1 (Principle of K-Theoretic Duality): $\psi = \psi(\psi) \implies \text{Topology of } G \text{ determines analysis of } C_r^*(G)$

The Baum-Connes Conjecture embodies:

• Groups know their operator algebras
• Topological invariants determine analytical invariants
• The assembly map is perfect self-knowledge
• This is ψ (group) recognizing ψ(ψ) (its operator algebra)

15.7 Known Cases

Theorem 15.1 (Proven Cases): Baum-Connes holds for:

1. Amenable groups (Higson-Kasparov)
2. Groups with Haagerup property
3. Hyperbolic groups (Mineyev-Yu)
4. Groups acting on trees
5. Many arithmetic groups

Theorem 15.2 (Inheritance): If G₁, G₂ satisfy BC, then so do:

• G₁ × G₂
• Subgroups of G₁
• Certain extensions

15.8 The Coarse Baum-Connes

Variant: For metric spaces instead of groups.

Definition 15.8 (Coarse Assembly): $\mu_X: KX_*(X) \to K_*(C^*(X))$

where:

• KX_* = coarse K-homology
• C*(X) = Roe algebra of X

Status: Counterexamples exist (Higson-Lafforgue-Skandalis)!

15.9 Consequences of Baum-Connes

If true for G, then:

Theorem 15.3 (Novikov Conjecture): Higher signatures are homotopy invariant.

Theorem 15.4 (Kadison-Kaplansky): No non-trivial idempotents in C[G].

Theorem 15.5 (Stable Gromov-Lawson-Rosenberg): Criteria for positive scalar curvature metrics.

15.10 The Analytical Side

Key Objects: K-theory of C_r^*(G) encodes:

• Representations of G
• Induced representations
• Elliptic operators on G-spaces

Computation: Generally very difficult!

15.11 The Topological Side

Definition 15.9 (Equivariant K-Homology): $K_*^G(EG) = \lim_{\text{compact } K \subset EG} K_*^G(K)$

Computation: Often more tractable using:

• Spectral sequences
• Chern character
• Induction methods

15.12 Expanders and Counterexamples

Warning: Related conjectures can fail!

Theorem 15.6 (Gromov-Lawson): Certain expander sequences provide counterexamples to coarse BC.

Mystery: Why does BC hold for groups but fail for spaces?

15.13 The Trace Conjecture

Related Problem: When is the trace on C_r^*(G) the only trace?

Conjecture 15.2 (Kadison-Kaplansky Trace): For torsion-free G, C_r^*(G) has unique normalized trace.

Connection: Follows from rational injectivity of BC assembly.

15.14 Computational Methods

Algorithm 15.1 (K-Theory Computation):

def compute_K_theory(G):
    # Topological side
    EG = classifying_space_proper(G)
    K_top = equivariant_K_homology(G, EG)
    
    # Analytical side (harder!)
    C_star = reduced_C_star_algebra(G)
    K_an = operator_K_theory(C_star)
    
    # Check if assembly is isomorphism
    assembly = assembly_map(K_top, K_an)
    
    return is_isomorphism(assembly)

Challenge: Both sides are infinite-dimensional!

15.15 Physics Connections

Applications to Physics:

1. Topological phases: K-theory classifies topological insulators
2. Index theory: Anomalies in quantum field theory
3. Noncommutative geometry: Quantum spaces

BC relates topological and analytical aspects of quantum systems.

15.16 The Farrell-Jones Alternative

Alternative Approach: Different assembly map.

Conjecture 15.3 (Farrell-Jones): Assembly map for algebraic K-theory and L-theory.

Relation: Similar philosophy, different target.

15.17 Geometric Group Theory

Impact on Geometric Group Theory:

• New invariants of groups
• Motivation for property (T), Haagerup property
• Connection to quasi-isometry invariants

BC drives research in group geometry.

15.18 Partial Results

Theorem 15.7 (With Coefficients): BC with coefficients holds for larger classes.

Strategy:

1. Prove BC with coefficients
2. Use permanence properties
3. Deduce rational BC
4. Hope for integral BC

15.19 Why It's Hard

Obstacles:

1. Two infinities: Both sides involve limits
2. Non-functoriality: Assembly isn't functorial
3. Lack of geometry: Abstract operator algebras
4. Counterexamples nearby: Coarse version fails

Each obstacle requires new techniques.

15.20 The Fifteenth Echo

The Baum-Connes Conjecture represents a deep test of ψ = ψ(ψ):

• Can groups recognize themselves through operator algebras?
• Does topology determine analysis completely?
• Is the assembly map perfect knowledge?

This conjecture claims that a group's topological nature (how it acts on spaces) completely determines its analytical nature (its operator algebra). It's a profound statement that discrete group structure creates continuous operator structure in a perfectly predictable way.

Whether true or false, the Baum-Connes Conjecture illuminates the mysterious relationship between the discrete and continuous, between groups and their operator algebras, between topology and analysis.

Each group whispers through its assembly map: "My topology knows my analysis, my proper actions determine my operator algebra, my K-theory is unified—for ψ = ψ(ψ) means that groups achieve perfect self-knowledge through the bridge between discrete and continuous."