Topology and Geometry for Physicists

Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. Nash & Sen - Topology and Geometry for Physicists Preface 1 - Basic Notions of Topology and the Value of Topological Reasoning 1.1. Introduction 1.2. Basic topological notions 1.3. Homeomorphisms, homotopy and the idea of topological invariants 1.4. Topological invariants of compactness and connectedness 1.5. Invariance of the dimension of R^n 2 - Differential Geometry: Manifolds and Differential Forms 2.1. Manifolds 2.2. Orientability 2.3. Calculus on manifolds 2.4. Infinite dimensional manifolds 2.5. Differentiable structures 3 - The Fundamental Group 3.1. Introduction 3.2. Definition of the fundamental group 3.3. Simplexes and the calculating theorem 3.4. Triangulation of a space with examples 3.4. Fundamental group of a product XxY 4 - The Homology Groups 4.1. Introduction 4.2. Oriented simplexes and the definition of the homology groups 4.3. Abelian groups 4.4. Relative homology groups 4.5. Exact sequences 4.6. Torsion, Kunneth formula, Euler-Poincaré formula and singular homology 5 - The Higher Homotopy Groups 5.1. Introduction 5.2. Definition of higher homotopy groups 5.3. Abelian nature of higher homotopy groups 5.4. Relative homotopy groups 5.5. The exact homotopy sequence 6 - Cohomology and De Rham Cohomology 6.1. Introduction 6.2. H^p(M;R) and Poincaré’s lemma 6.3. Poincaré’s lemma 6.4. Calculation of H^p(M;R) 6.5. General remarks 6.6. The cup product 6.7. Superiority of cohomology over homology 7 - Fibre Bundles and Further Differential Geometry 7.1. Introduction 7.2. Fibre bundle 7.3. More examples of bundles 7.4. When is a bundle trivial? 7.5. Sections of bundles and singularities of vector fields 7.6. Cutting a bundle down to size: reduction of the group and contraction of the base space 7.7. Remarks on almost Hamiltonian and almost complex Structures 7.8. G-structures on a compact closed manifold M 7.9. Lie derivative 7.10. Connection and curvature 7.11. The connection form and the gauge potential 7.12. Parallel transport, covariant derivative and curvature 7.13. Covariant exterior derivatives 7.14. The Bianchi identities and *F 7.15. Connection in the tangent bundle 7.16. The torsion tensor 7.17. Geodesics 7.18. The Levi-Civita connection 7.19. The Yang-Mills connection 7.20. The Maxwell connection 7.21. General remarks 7.22. Characteristic classes 7.23. Chern, Pontrjagin and Euler classes 7.24. Characteristic classes in terms of curvature and invariant polynomials 7.25. Classification of bundles 7.26. The Stiefel-Whitney class 7.27. Calculation of characteristic classes 7.28. General remarks 7.29 Formulae obeyed by characteristic classes 7.30 Global invariants and local geometry 8 - Morse Theory 8.1. More inequalities 8.2. Morse lemma 8.3. Symmetry breaking selection rules in crystals 8.4. Estimating equilibrium positions 9 - Defects, Textures and Homotopy Theory 9.1. Planar spin in two dimensions 9.2. Definition of an ordered medium 9.3. Stability of defects theorem 9.4. Examples 9.5. General remarks and crossing of defects, textures and π_3(S^2) 10 - Yang-Mills Theories: Instantons and Monopoles 10.1. Introduction 10.2. Instantons 10.3. Topology and boundary conditions 10.4. Instantons and absolute minima 10.5. The instanton solution 10.6. The instanton number and the second Chern class 10,7. Multi-instantons 10.8. Quaternions and SU(2) connections 10.9. The k=1 instanton in terms of quaternions 10.10. Instantons with |k|>1 and quaternions 10.11. Example of instantons with |k|>1 10.12. Twistor methods and instantons 10.13. The projective twistor space 10.14. Twistor space and planes in C^4 10.15 α-planes and anti-self-dual connections 10.16 The equivalence between instantons and holomorphic vector bundles 10.17 Construction of an instanton given a holomorphic vector bundle 10.18 The Minkowski case 10.19 Monopoles 10.20. The Bohm-Aharanov effect Further Reading Subject Index