This volume studies topological spaces through algebraic invariants. It translates geometric and continuous structure into groups, rings, modules, and homological data.
Part I. Fundamental Constructions
Chapter 1. Spaces and Maps
1.1 Topological spaces 1.2 Continuous maps 1.3 Homotopy 1.4 Homotopy equivalence 1.5 Examples
Chapter 2. Fundamental Group
2.1 Paths and loops 2.2 Homotopy classes 2.3 Definition of π₁ 2.4 Functoriality 2.5 Examples
Chapter 3. Covering Spaces
3.1 Covering maps 3.2 Lifting properties 3.3 Universal covers 3.4 Deck transformations 3.5 Applications
Part II. Homology
Chapter 4. Simplicial Homology
4.1 Simplicial complexes 4.2 Chains and boundaries 4.3 Cycles and homology groups 4.4 Computations 4.5 Examples
Chapter 5. Singular Homology
5.1 Singular simplices 5.2 Chain complexes 5.3 Homotopy invariance 5.4 Exact sequences 5.5 Examples
Chapter 6. Homology Computations
6.1 Mayer-Vietoris sequence 6.2 Relative homology 6.3 Excision theorem 6.4 Degree theory 6.5 Applications
Part III. Cohomology
Chapter 7. Cohomology Groups
7.1 Cochains 7.2 Coboundary maps 7.3 Cohomology groups 7.4 Universal coefficient theorem 7.5 Examples
Chapter 8. Cup Products
8.1 Ring structure 8.2 Graded commutativity 8.3 Computations 8.4 Applications 8.5 Examples
Chapter 9. Poincare Duality
9.1 Orientable manifolds 9.2 Fundamental class 9.3 Duality statement 9.4 Applications 9.5 Examples
Part IV. Homotopy Theory
Chapter 10. Higher Homotopy Groups
10.1 Definitions 10.2 Functoriality 10.3 Relation to π₁ 10.4 Computation difficulties 10.5 Examples
Chapter 11. Fiber Bundles and Fibrations
11.1 Fiber bundles 11.2 Fibrations 11.3 Long exact sequence of homotopy 11.4 Applications 11.5 Examples
Chapter 12. Spectra and Stable Homotopy
12.1 Suspension 12.2 Stable phenomena 12.3 Spectra overview 12.4 Generalized cohomology 12.5 Applications
Part V. Characteristic Classes
Chapter 13. Vector Bundles
13.1 Definitions 13.2 Bundle maps 13.3 Operations on bundles 13.4 Classification 13.5 Examples
Chapter 14. Characteristic Classes
14.1 Stiefel-Whitney classes 14.2 Chern classes 14.3 Euler class 14.4 Pontryagin classes 14.5 Applications
Chapter 15. K-Theory Connections
15.1 Vector bundles and K-theory 15.2 Bott periodicity overview 15.3 Cohomology operations 15.4 Applications 15.5 Examples
Part VI. Advanced Methods
Chapter 16. Spectral Sequences
16.1 Filtrations 16.2 Exact couples 16.3 Serre spectral sequence 16.4 Adams spectral sequence overview 16.5 Applications
Chapter 17. Cohomology Operations
17.1 Steenrod squares 17.2 Reduced powers 17.3 Operations and structure 17.4 Applications 17.5 Examples
Chapter 18. Homological Algebra in Topology
18.1 Chain complexes 18.2 Derived functors 18.3 Ext and Tor 18.4 Universal coefficient theorems 18.5 Applications
Part VII. Applications
Chapter 19. Manifolds
19.1 Classification invariants 19.2 Intersection forms 19.3 Cobordism overview 19.4 Surgery theory overview 19.5 Applications
Chapter 20. Fixed Point Theory
20.1 Brouwer fixed point theorem 20.2 Lefschetz fixed point theorem 20.3 Index methods 20.4 Applications 20.5 Examples
Chapter 21. Applied and Computational Topology
21.1 Persistent homology 21.2 Simplicial data models 21.3 Topological data analysis 21.4 Algorithms 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Homotopy type theory links 22.2 Higher category theory 22.3 Chromatic homotopy theory overview 22.4 Equivariant homotopy theory 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Homotopy group computations 23.2 Manifold classification 23.3 Stable homotopy questions 23.4 Computational complexity 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of algebraic topology 24.2 Key contributors 24.3 Evolution of invariants 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard spaces and invariants B. Exact sequence reference C. Proof techniques checklist D. Computation templates E. Cross-reference to other MSC branches