Preface |
Basic Notions of Topology and the Value of Topological Reasoning / Chapter 1: |
Introduction / 1.1: |
Basic topological notions / 1.2: |
Homeomorphisms, homotopy and the idea of topological invariants / 1.3: |
Topological invariants of compactness and connectedness / 1.4: |
Differential Geometry: Manifolds and Differential Forms / 1.5: |
Manifolds / 2.1: |
Orientability / 2.2: |
Calculus on manifolds / 2.3: |
Infinite dimensional manifolds / 2.4: |
Differentiable structures / 2.5: |
The Fundamental Group / Chapter 3: |
Definition of the fundamental group / 3.1: |
Simplexes and the calculating theorem / 3.3: |
Triangulation of a space with examples / 3.4: |
Fundamental group of a product X×Y / 3.5: |
The Homology Groups / Chapter 4: |
Oriented simplexes and the definition of the homology groups / 4.1: |
Abelian groups / 4.3: |
Relative homology groups / 4.4: |
Exact sequences / 4.5: |
Torsion, Kunneth formula, Euler-Poincaré formula and singular homology / 4.6: |
The Higher Homotopy Groups / Chapter 5: |
Definition of higher homotopy groups / 5.1: |
Abelian nature of higher homotopy groups / 5.3: |
Relative homotopy groups / 5.4: |
The exact homotopy sequence / 5.5: |
Cohomology and De Rham Cohomology / Chapter 6: |
Poincaré's lemma / 6.1: |
General remarks / 6.4: |
The cup product / 6.6: |
Superiority of cohomology over homology / 6.7: |
Fibre Bundles and Further Differential Geometry / Chapter 7: |
Fibre bundle / 7.1: |
More examples of bundles / 7.3: |
When is a bundle trivial? / 7.4: |
Sections of bundles and singularities of vector fields / 7.5: |
Cutting a bundle down to size: reduction of the group and contraction of the base space / 7.6: |
Remarks on almost Hamiltonian and almost complex structures / 7.7: |
G-structures on a compact closed manifold M / 7.8: |
Lie derivative / 7.9: |
Connection and curvature / 7.10: |
The connection form and the gauge potential / 7.11: |
Parallel transport, covariant derivative and curvature / 7.12: |
Covariant exterior derivatives / 7.13: |
The Bianchi identities and *F / 7.14: |
Connection in the tangent bundle / 7.15: |
The torsion tensor / 7.16: |
Geodesics / 7.17: |
The Levi-Civita connection / 7.18: |
The Yang-Mills connection / 7.19: |
The Maxwell connection / 7.20: |
Characteristic classes / 7.21: |
Chern, Pontrjagin and Euler classes / 7.23: |
Characteristic classes in terms of curvature and invariant polynomials / 7.24: |
Classification of bundles / 7.25: |
The Stiefel-Whitney class / 7.26: |
Calculation of characteristic classes / 7.27: |
Formulae obeyed by characteristic classes / 7.28: |
Global invariants and local geometry / 7.30: |
Morse Theory / Chapter 8: |
More inequalities / 8.1: |
Morse lemma / 8.2: |
Symmetry breaking selection rules in crystals / 8.3: |
Estimating equilibrium positions / 8.4: |
Defects, Textures and Homotopy Theory / Chapter 9: |
Planar spin in two dimensions / 9.1: |
Definition of an ordered medium / 9.2: |
Stability of defects theorem / 9.3: |
Examples / 9.4: |
Yang-Mills Theories: Instantons and Monopoles / 9.5: |
Instantons / 10.1: |
Topology and boundary conditions / 10.3: |
Instantons and absolute minima / 10.4: |
The instanton solution / 10.5: |
The instanton number and the second Chern class / 10.6: |
Multi-instantons / 10.7: |
Quaternions and SU(2) connections / 10.8: |
The k = 1 instanton in terms of quaternions / 10.9: |
Instantons with |k|> 1 and quaternions / 10.10: |
Example of instantons with |k|> 1 / 10.11: |
Twistor methods and instantons / 10.12: |
The projective twistor space / 10.13: |
?-planes and anti-self-dual connections / 10.14: |
The equivalence between instantons and holomorphic vector bundles / 10.16: |
Construction of an instanton given a holomorphic vector bundle / 10.17: |
The Minkowski case / 10.18: |
Monopoles / 10.19: |
The Bohm-Aharanov effect / 10.20: |
Further Reading |
Subject Index |
Preface |
Basic Notions of Topology and the Value of Topological Reasoning / Chapter 1: |
Introduction / 1.1: |
Basic topological notions / 1.2: |
Homeomorphisms, homotopy and the idea of topological invariants / 1.3: |
Topological invariants of compactness and connectedness / 1.4: |