This volume studies manifolds and combinatorial models such as CW complexes.
This volume studies manifolds and combinatorial models such as CW complexes. It connects topology, geometry, and algebraic topology through structured spaces.
Part I. Topological Manifolds
Chapter 1. Definitions and Examples
1.1 Topological manifolds 1.2 Local Euclidean structure 1.3 Charts and atlases 1.4 Examples 1.5 Basic properties
Chapter 2. Maps Between Manifolds
2.1 Continuous maps 2.2 Homeomorphisms 2.3 Embeddings 2.4 Submanifolds 2.5 Examples
Chapter 3. Topological Properties
3.1 Compactness 3.2 Connectedness 3.3 Orientability 3.4 Boundary 3.5 Examples
Part II. Smooth Manifolds
Chapter 4. Smooth Structures
4.1 Smooth atlases 4.2 Compatibility 4.3 Smooth manifolds 4.4 Examples 4.5 Constructions
Chapter 5. Smooth Maps
5.1 Differentiable maps 5.2 Immersions and submersions 5.3 Diffeomorphisms 5.4 Examples 5.5 Applications
Chapter 6. Tangent Spaces and Bundles
6.1 Tangent vectors 6.2 Tangent bundle 6.3 Vector fields 6.4 Examples 6.5 Applications
Part III. CW Complexes
Chapter 7. Cell Complexes
7.1 Definitions 7.2 Attaching cells 7.3 Skeleta 7.4 Examples 7.5 Basic properties
Chapter 8. Homotopy Properties
8.1 Cellular maps 8.2 Homotopy equivalence 8.3 Cellular approximation theorem 8.4 Applications 8.5 Examples
Chapter 9. Homology of CW Complexes
9.1 Cellular homology 9.2 Computations 9.3 Applications 9.4 Examples 9.5 Connections
Part IV. Intersection of Manifolds and CW Complexes
Chapter 10. Triangulations
10.1 Simplicial complexes 10.2 Triangulating manifolds 10.3 Compatibility 10.4 Applications 10.5 Examples
Chapter 11. Handle Decompositions
11.1 Handles and attachments 11.2 Morse functions overview 11.3 Decomposition of manifolds 11.4 Applications 11.5 Examples
Chapter 12. Morse Theory (Overview)
12.1 Critical points 12.2 Morse inequalities 12.3 Topological consequences 12.4 Applications 12.5 Examples
Part V. Fiber Bundles and Structures
Chapter 13. Fiber Bundles
13.1 Definitions 13.2 Local triviality 13.3 Examples 13.4 Structure groups 13.5 Applications
Chapter 14. Vector Bundles
14.1 Definitions 14.2 Sections 14.3 Bundle maps 14.4 Examples 14.5 Applications
Chapter 15. Principal Bundles
15.1 Definitions 15.2 Group actions 15.3 Connections overview 15.4 Applications 15.5 Examples
Part VI. Differential and Geometric Structures
Chapter 16. Differential Forms
16.1 Exterior algebra 16.2 Differential forms 16.3 Integration 16.4 Stokes theorem 16.5 Applications
Chapter 17. Riemannian Structures
17.1 Metrics 17.2 Geodesics 17.3 Curvature overview 17.4 Applications 17.5 Examples
Chapter 18. Symplectic and Complex Structures (Overview)
18.1 Symplectic manifolds 18.2 Complex manifolds 18.3 Compatibility conditions 18.4 Applications 18.5 Connections
Part VII. Applications
Chapter 19. Algebraic Topology Connections
19.1 Homotopy and homology 19.2 Characteristic classes 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Geometry Applications
20.1 Classification of manifolds 20.2 Geometric structures 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Computational Topology
21.1 Meshes and triangulations 21.2 Algorithms 21.3 Data representations 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 High-dimensional manifolds 22.2 Surgery theory overview 22.3 Exotic structures 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification of manifolds 23.2 Smooth structure questions 23.3 High-dimensional topology 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of manifold theory 24.2 Key contributors 24.3 Evolution of cell complexes 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard manifold examples B. CW complex constructions C. Proof techniques checklist D. Decomposition methods E. Cross-reference to other MSC branches