Modern Geometry-Methods and Applications

Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics. The three volumes of Modern Geometry - Methods and Applications contain a concrete exposition of these methods together with their main applications in mathematics and physics. This third volume, presented in highly accessible languages, concentrates in homology theory. It contains introductions to the contemporary methods for the calculation of homology groups and the classification of manifesto. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

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.