This volume studies algebraic and topological K-theory, focusing on invariants derived from vector bundles, modules, and operator algebras.
This volume studies algebraic and topological K-theory, focusing on invariants derived from vector bundles, modules, and operator algebras. It provides tools for classification problems across algebra, topology, and geometry.
Part I. Foundations of K-Theory
Chapter 1. Motivation and Basic Definitions
1.1 Classification problems 1.2 Grothendieck groups 1.3 Additive invariants 1.4 Exact sequences 1.5 Examples
Chapter 2. Grothendieck Construction
2.1 Monoids to groups 2.2 Universal properties 2.3 Functoriality 2.4 Examples in algebra and topology 2.5 Basic properties
Chapter 3. Exact Categories
3.1 Exact sequences 3.2 Additive categories 3.3 Exact functors 3.4 Examples 3.5 Structural properties
Part II. Algebraic K-Theory
Chapter 4. K₀ of Rings
4.1 Projective modules 4.2 Definition of K₀ 4.3 Computation examples 4.4 Functoriality 4.5 Applications
Chapter 5. Higher Algebraic K-Theory
5.1 Definition of K₁ 5.2 General linear groups 5.3 Definition of K₂ (overview) 5.4 Higher K-groups (overview) 5.5 Applications
Chapter 6. Localization and Excision
6.1 Exact sequences in K-theory 6.2 Localization sequences 6.3 Excision properties 6.4 Applications 6.5 Examples
Part III. Topological K-Theory
Chapter 7. Vector Bundles
7.1 Definitions 7.2 Operations on bundles 7.3 Classification 7.4 Examples 7.5 Applications
Chapter 8. Topological K-Groups
8.1 Definition of K⁰ 8.2 Reduced K-theory 8.3 Suspension and Bott periodicity 8.4 Computation examples 8.5 Applications
Chapter 9. Bott Periodicity
9.1 Statement and consequences 9.2 Loop spaces 9.3 Periodicity phenomena 9.4 Applications 9.5 Examples
Part IV. Connections to Geometry
Chapter 10. K-Theory and Cohomology
10.1 Chern classes 10.2 Characteristic classes 10.3 Riemann–Roch theorem (overview) 10.4 Relations to cohomology 10.5 Applications
Chapter 11. Index Theory
11.1 Elliptic operators 11.2 Analytical index 11.3 Topological index 11.4 Index theorem (overview) 11.5 Applications
Chapter 12. K-Theory of Schemes
12.1 Algebraic vector bundles 12.2 K-theory of varieties 12.3 Relations to algebraic geometry 12.4 Examples 12.5 Applications
Part V. Operator Algebras and K-Theory
Chapter 13. C*-Algebras
13.1 Definitions 13.2 Representations 13.3 Examples 13.4 Structure theory 13.5 Applications
Chapter 14. K-Theory for Operator Algebras
14.1 K₀ and K₁ of C*-algebras 14.2 Exact sequences 14.3 Computation methods 14.4 Applications 14.5 Examples
Chapter 15. Noncommutative Geometry
15.1 Basic ideas 15.2 K-theory as invariant 15.3 Cyclic cohomology (overview) 15.4 Applications 15.5 Connections
Part VI. Advanced Topics
Chapter 16. Higher Structures
16.1 Spectra (overview) 16.2 Stable homotopy perspective 16.3 Generalized cohomology 16.4 Applications 16.5 Examples
Chapter 17. Motivic K-Theory (Overview)
17.1 Definitions 17.2 Relations to algebraic geometry 17.3 Cohomological structures 17.4 Applications 17.5 Examples
Chapter 18. Equivariant K-Theory
18.1 Group actions 18.2 Equivariant bundles 18.3 Equivariant K-groups 18.4 Applications 18.5 Examples
Part VII. Computational and Applied Aspects
Chapter 19. Computation of K-Groups
19.1 Exact sequences 19.2 Spectral sequences 19.3 Known computations 19.4 Algorithmic approaches 19.5 Examples
Chapter 20. Applications
20.1 Topology 20.2 Geometry 20.3 Physics 20.4 Operator algebras 20.5 Data and computation
Chapter 21. Interdisciplinary Connections
21.1 Number theory links 21.2 Algebraic geometry links 21.3 Functional analysis links 21.4 Mathematical physics 21.5 Emerging areas
Part VIII. Research Directions
Chapter 22. Open Problems
22.1 Computation challenges 22.2 Classification questions 22.3 Higher K-theory gaps 22.4 Structural questions 22.5 Future directions
Chapter 23. Emerging Areas
23.1 Derived algebraic geometry 23.2 Homotopical methods 23.3 Noncommutative spaces 23.4 Computational K-theory 23.5 Trends
Chapter 24. Historical and Conceptual Notes
24.1 Development of K-theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard constructions B. Key theorems summary C. Proof techniques checklist D. Computational tools E. Cross-reference to other MSC branches
This volume develops K-theory as a unifying invariant framework. It connects algebra, topology, and analysis through classification and structural insight.