Higher Index Theory

Contents
Introduction
PART 1: Background
	1 C*-algebras
		1.1 Definition and Examples
		1.2 Invertible Elements and Spectrum
		1.3 Commutative C*-algebras
		1.4 Functional Calculus
		1.5 Ideals and Quotients
		1.6 Spatial Theory
		1.7 Multipliers and Corners
		1.8 Tensor Products
		1.9 Exercises
		1.10 Notes and References
	2 K-theory for C*-algebras
		2.1 Algebraic K0
		2.2 Approximation and Homotopy in K0
		2.3 Unbounded Traces
		2.4 The Algebraic Index Map
		2.5 The Topological K1 Group
		2.6 Bott Periodicity and the Six-Term Exact Sequence
		2.7 Some Computational Tools
		2.8 Index Elements
		2.9 The Spectral Picture of K-theory
		2.10 The External Product in K-theory
		2.11 Exercises
		2.12 Notes and References
	3 Motivation: Positive Scalar Curvature on Tori
		3.1 Differential Geometry
		3.2 Hilbert Space Techniques
		3.3 K-theory Computations
		3.4 Some Historical Comments
		3.5 Content of This Book
		3.6 Exercises
PART 2: Roe Algebras, Localisation Algebras and Assembly
	4 Geometric Modules
		4.1 Geometric Modules
		4.2 Covering Isometries
		4.3 Covering Isometries for Coarse Maps
		4.4 Covering Isometries for Continuous Maps
		4.5 Equivariant Covering Isometries
		4.6 Exercises
		4.7 Notes and References
	5 Roe Algebras
		5.1 Roe Algebras
		5.2 Equivariant Roe Algebras
		5.3 Relationship to Group C∗-algebras
		5.4 Exercises
		5.5 Notes and References
	6 Localisation Algebras and K-homology
		6.1 Asymptotically Commuting Families
		6.2 Localisation Algebras
		6.3 K-homology
		6.4 General Functoriality
		6.5 Equivariant K-homology
		6.6 The Localised Roe Algebra
		6.7 Other Pictures of K-homology
		6.8 Exercises
		6.9 Notes and References
	7 Assembly Maps and the Baum–Connes Conjecture
		7.1 Assembly and the Baum–Connes Conjecture
		7.2 Rips Complexes
		7.3 Uniformly Contractible Spaces
		7.4 Classifying Spaces
		7.5 The Coarse Baum–Connes Conjecture for Euclidean Space
		7.6 Exercises
		7.7 Notes and References
PART 3: Differential Operators
	8 Elliptic Operators and K-homology
		8.1 Differential Operators and Self-Adjointness
		8.2 Wave Operators and Multipliers of L∗(M)
		8.3 Ellipticity and K-homology
		8.4 Schatten Classes
		8.5 Exercises
		8.6 Notes and References
	9 Products and Poincare´ Duality
		9.1 A Concrete Pairing between K-homology and K-theory
		9.2 General Pairings and Products
		9.3 The Dirac Operator on Rd and Bott Periodicity
		9.4 Representable K-homology
		9.5 The Cap Product
		9.6 The Dirac Operator on a Spinc Manifold and Poincare´ Duality
		9.7 Exercises
		9.8 Notes and References
	10 Applications to Algebra, Geometry and Topology
		10.1 The Kadison–Kaplansky Conjecture
		10.2 Positive Scalar Curvature and Secondary Invariants
		10.3 The Novikov Conjecture
		10.4 Exercises
		10.5 Notes and References
PART 4 Higher Index Theory and Assembly
	11 Almost Constant Bundles
		11.1 Pairings
		11.2 Non-positive Curvature
		11.3 Exercises
		11.4 Notes and References
	12 Higher Index Theory for Coarsely Embeddable Spaces
		12.1 The Bott–Dirac Operator
		12.2 Bounded Geometry Spaces
		12.3 Index Maps
		12.4 The Local Isomorphism
		12.5 Reduction to Coarse Disjoint Unions
		12.6 The Case of Coarse Disjoint Unions
		12.7 Exercises
		12.8 Notes and References
	13 Counterexamples
		13.1 Injectivity Counterexamples from Large Spheres
		13.2 Expanders and Property (τ)
		13.3 Surjectivity Counterexamples from Expanders
		13.4 Exercises
		13.5 Notes and References
APPENDICES
A Topological Spaces, Group Actions and Coarse Geometry
	A.1 Topological Spaces
	A.2 Group Actions on Topological Spaces
	A.3 Coarse Geometry
	A.4 Exercises
	A.5 Notes and References
B Categories of Topological Spaces and Homology Theories
	B.1 Categories We Work With
	B.2 Homology Theories on LC
	B.3 Exercises
	B.4 Notes and References
C Unitary Representations
	C.1 Unitary Representations
	C.2 Fell’s Trick
	C.3 Notes and References
D Unbounded Operators
	D.1 Self-Adjointness and the Spectral Theorem
	D.2 Some Fourier Theory for Unbounded Operators
	D.3 The Harmonic Oscillator and Mehler’s Formula
	D.4 Notes and References
E Gradings
	E.1 Graded C*-algebras and Hilbert Spaces
	E.2 Graded Tensor Products
	E.3 Exercises
	E.4 Notes and References
References
Index of Symbols
Index