- 现代几何学方法和应用(第3卷英文版)
- - 作者：B.A.Dubrovin//A.T.Fomenko//S.P.Novikov|责编:刘慧//高蓉
  - 出版社：世界图书出版公司
  - ISBN：9787506212649
  - 出版日期：1999/11/01
  - 页数：416
- 售价：39.6

内容大纲
本书系统阐述了同调理论的基本知识，自从庞加莱奠定了拓扑学的基础之后，同调理论就被认为是学习代数拓扑学的基本入门知识，因此，本书对于广大研究生学好同调理论并进而研究拓学都是一本极好的教材。
作者介绍
目录
Preface
CHAPTER ⅠHomology 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. Poincar 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 n  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 nontri'ial 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. 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 Ω(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
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 facts about eobordisms.The signature
  27.2. Thorn complexes. Calculation of cobordisms (modulo torsion).The signature formula, Realization of
  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 II

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