Over the past 15 years, the geometrical and topological methods of the theory of manifolds have assumed 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 application in mathematics and physics. This third volume, presented in highly accessible language, concentrates on homology theory. It contains introduction to the contemporary methods for the calculation of homotopy groups and the classification of manifolds. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

Preface

**CHAPTER 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. Poincare duality in simplicial homology

7. The homology groups of a product of spaces. Multiplication in cohomology rings. The cohomology theory of H-spaces and Liegroups. The eo homology of the unitary groups

8. The homology theory offibre 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 p

_{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 non trivial 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-gapotentials

13. The simplest properties of Kahler manifolds. Abelian tori

14. Sheaf cohomology

**CHAPTER 2**

Critical Points of Smooth Functions and Homology Theory

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. Poincare 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 O(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 nobody problem

**CHAPTER 3**

Cobordisms and Smooth Structures

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. Thorn 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

Bibliography

APPENDIX 1 (by S. P. Novikov)

An Analogue of Morse Theory for Many-Valued Functions. Certain Properties of Poisson Brackets

APPENDIX 2 (by A. T. Fomenko)

Plateau's Problem. Spectral Bordisms and Globally Minimal Surfaces in Riemannian Manifolds

Index

Errata to Parts I and 11