Modern Geometry Methods and Applications: Part 3: Introduction to Homology Theory

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.-
13. The simplest properties of Kähler manifolds. Abelian tori.-
14. Sheaf cohomology.- 2 Critical Points of Smooth Functions and Homology Theory.-
15. Morse functions and cell complexes.-
16. The Morse inequalities.-
17. Morse-Smale functions. Handles. Surfaces.-
18. Poincaré duality.-
19. Critical points of smooth functions and the Lyusternik-Shnirelman category of a manifold.-
20. Critical manifolds and the Morse inequalities. Functions with symmetry.-
21. Critical points of functionals and the topology of the path space ?(M).-
22. Applications of the index theorem.-
23. The periodic problem of the calculus of variations.-
24. Morse functions on 3-dimensional manifolds and Heegaard splittings.- 25. Unitary Bott periodicity and higher-dimensional variational problems.- 25.1. The theorem on unitary periodicity.- 25.2. Unitary periodicity via the two-dimensional calculus of variations.- 25.3. Onthogonal periodicity via the higher-dimensional calculus of variations.-
26. Morse theory and certain motions in the planar n-body problem.- 3 Cobordisms and Smooth Structures.-
27. Characteristic numbers. Cobordisms. Cycles and submanifolds. The signature of a manifold.- 27.1. Statement of the problem. The simplest facts about cobordisms. The signature.- 27.2. Thom complexes. Calculation of cobordisms (modulo torsion). The signature formula. Realization of cycles as submanifolds.- 27.3. Some applications of the signature formula. The signature and the problem of the invariance of classes.-
28. Smooth structures on the 7-dimensional sphere. The classification problem for smooth manifolds (normal invariants). Reidemeister torsion and the fundamental hypothesis (Hauptvermutung) of combinatorial topology.- APPENDIX 1 An Analogue of Morse Theory for Many-Valued Functions. Certain Properties of Poisson Brackets.- APPENDIX 2 Plateau's Problem. Spectral Bordisms and Globally Minimal Surfaces in Riemannian Manifolds.- Errata to Parts I and II.