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. - 10. Homology theory and methods for computing homotopy groups. The Cartan-Serre theorem. Cohomology operations. Vector bundles. - 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 theLyusternik-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. - 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. - 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. Language: English