Modern Geometry Methods and Applications

1 Homology and Cohomology. Computational Recipes
- §1. Cohomology groups as classes of closed differential forms. Their homotopy invariance
- §2. The homology theory of algebraic complexes
- §3. Simplicial complexes. Their homology and cohomology groups. The classification of the two-dimensional closed surfaces
- §4. Attaching cells to a topological space. Cell spaces. Theorems on the reduction of cell spaces. Homology groups and the fundamental groups of surfaces and certain other manifolds
- §5. The singular homology and cohomology groups. Their homotogy invariance. The exact sequence of a pair. Relative homology groups
- §6. The singular homology of cell complexes. Its equivalence with cell homology. Poincaré duality in simplicial homology
- §7. The homology groups of a product of spaces. Multiplication in cohomology rings. The cohomology theory of H-spaces and Lie groups. The cohomology of the unitary groups
- §8. The homology theory of fibre bundles (skew products)
- §9. The extension problem for maps, homotopies, and cross-sections. Obstruction cohomology classes
9.1. The extension problem for maps
9.2. The extension problem for homotopies
9.3. The extension problem for cross-sections
- §10. Homology theory and methods for computing homotopy groups. The Cartan-Serre theorem. Cohomology operations. Vector bundles
10.1. The concept of a cohomology operation. Examples
10.2. Cohomology operations and Eilenberg-MacLane complexes
10.3. Computation of the rational homotopy groups ?i ? Q
10.4. Application to vector bundles. Characteristic classes
10.5. Classification of the Steenrod operations in low dimensions
10.6. Computation of the first few nontrivial stable homotopy groups of spheres
10.7. Stable homotopy classes of maps of cell complexes
- §11. Homology theory and the fundamental group
- §12. The cohomology groups of hyperelliptic Riemann surfaces. Jacobi tori. Geodesics on multi-axis ellipsoids. Relationship to finite-gap potentials
-