1
Scope, conventions, and prerequisites
▶
1.1
The version of the conjecture
1.2
Standing assumptions
1.3
Background interfaces
1.4
Status conventions
2
The reduced group \(C^*\)-algebra
▶
2.1
The complex group algebra
2.2
The Hilbert space and left translations
2.3
The integrated representation
2.4
The reduced norm and operator-norm closure
2.5
Countability and a first calibration
2.6
Interface for subsequent parts
3
Operator K-theory and the analytic side
▶
3.1
Projections and Murray–von Neumann equivalence
3.2
Matrix stabilization and the construction of \(K_0\)
3.3
Functoriality of unital \(K_0\)
3.4
Invariance under unital star-algebra isomorphism
3.5
Unitaries, norm homotopy, and \(K_1\)
3.6
Nonunital algebras and structural properties
3.7
The analytic target
4
Proper actions and the universal proper space
▶
4.1
Proper actions and compact transporters
4.2
Subgroup fixed-point spaces
4.3
Equivariant maps and homotopies
4.4
Homogeneous orbits and orbit cells
4.5
The fixed-point criterion and uniqueness
4.6
Cocompact indexing for a given ambient space
4.7
Categorical equivariant cell attachments
4.8
A concrete probability target
4.9
Closed skeletal inclusions and the colimit topology
4.10
Real-valued extension and Hausdorffness
4.11
Constructed zero-skeleta and finite-group CW models
4.12
Labeled orbit simplices for a join construction
4.13
Equivariant CW complexes and a model for \(\underline E\Gamma \)
4.14
Cocompact pieces and model choices
5
Equivariant KK-theory
▶
5.1
Hilbert modules and operator ideals
5.2
Equivariant Kasparov cycles
5.3
Kasparov products and functoriality
5.4
Comparison with operator K-theory
6
The topological side
▶
6.1
K-homology of a cocompact piece
6.2
Transition maps
6.3
The filtered colimit
7
Construction of the assembly map
▶
7.1
Reduced crossed products and descent
7.2
Integrated source action and genuine quotient descent
7.3
Cutoff projections
7.4
Naturality for actual coefficient and cutoff maps
7.5
Assembly on a piece
7.6
Transitions and the actual colimit
8
The conjecture, special cases, and status
▶
8.1
The property for one group
8.2
The general conjecture
8.3
Calibration examples and later specializations
8.4
Constructions, comparisons, and the scope of the conjecture
8.5
Research and formalization status
Chapter 1 graph
Chapter 2 graph
Chapter 3 graph
Chapter 4 graph
Chapter 5 graph
Chapter 6 graph
Chapter 7 graph
Chapter 8 graph
Baum-Connes Conjecture
1
Scope, conventions, and prerequisites
1.1
The version of the conjecture
1.2
Standing assumptions
1.3
Background interfaces
1.4
Status conventions
2
The reduced group \(C^*\)-algebra
2.1
The complex group algebra
2.2
The Hilbert space and left translations
2.3
The integrated representation
2.4
The reduced norm and operator-norm closure
2.5
Countability and a first calibration
2.6
Interface for subsequent parts
3
Operator K-theory and the analytic side
3.1
Projections and Murray–von Neumann equivalence
3.2
Matrix stabilization and the construction of \(K_0\)
3.3
Functoriality of unital \(K_0\)
3.4
Invariance under unital star-algebra isomorphism
3.5
Unitaries, norm homotopy, and \(K_1\)
3.6
Nonunital algebras and structural properties
3.7
The analytic target
4
Proper actions and the universal proper space
4.1
Proper actions and compact transporters
4.2
Subgroup fixed-point spaces
4.3
Equivariant maps and homotopies
4.4
Homogeneous orbits and orbit cells
4.5
The fixed-point criterion and uniqueness
4.6
Cocompact indexing for a given ambient space
4.7
Categorical equivariant cell attachments
4.8
A concrete probability target
4.9
Closed skeletal inclusions and the colimit topology
4.10
Real-valued extension and Hausdorffness
4.11
Constructed zero-skeleta and finite-group CW models
4.12
Labeled orbit simplices for a join construction
4.13
Equivariant CW complexes and a model for \(\underline E\Gamma \)
4.14
Cocompact pieces and model choices
5
Equivariant KK-theory
5.1
Hilbert modules and operator ideals
5.2
Equivariant Kasparov cycles
5.3
Kasparov products and functoriality
5.4
Comparison with operator K-theory
6
The topological side
6.1
K-homology of a cocompact piece
6.2
Transition maps
6.3
The filtered colimit
7
Construction of the assembly map
7.1
Reduced crossed products and descent
7.2
Integrated source action and genuine quotient descent
7.3
Cutoff projections
7.4
Naturality for actual coefficient and cutoff maps
7.5
Assembly on a piece
7.6
Transitions and the actual colimit
8
The conjecture, special cases, and status
8.1
The property for one group
8.2
The general conjecture
8.3
Calibration examples and later specializations
8.4
Constructions, comparisons, and the scope of the conjecture
8.5
Research and formalization status