本书是莫斯科大学数学力学系经典教材《现代几何学——方法和应用》三卷本的第3卷。全书力求以直观的和物理的视角阐述，是一本难得的现代几何方面的佳作。整套书内容包括张量分析、曲线和曲面几何、一维和高维变分法（第1卷），微分流形的拓扑和几何（第2卷），以及同调与上同调理论（第3卷）。本书可用作数学和理论物理专业高年级和研究生的教学用书，对从事几何和拓扑研究的工作者也极具参考价值。

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 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 πi □(特殊字符)
10.4. Application to vector bundles. Characterist ic 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 Kihler manifolds. Abelian tori
14. Sheaf cohomology
CHAPTER 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. Poinoare duality
19. Critical points of smooth functions and the Lyusternik- Shnirelman category of a manifold
20. Critical manifolds and the Morse inequalities. Fuactions with symmetry
21. Critical points of functionals and the topology of the path space Ω(M)
22. Applications of the index theorem
23. The periodic problem of the calculus of variation
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-dimensiona, calculus of variations
25.3. Onthogonal periodicity via the higher-dime asional calculus of variations
26. Morse theory and certain motions in the planar n-body problem
CHAPTER 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 fact,; 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.
Cer