現代幾何學方法和應用‧第3卷（簡體書）

In expositions of the elements of topology it is customary for homology to be given a fundamental role. Since Poincare, who laid the foundations of topology, homology theory has been regarded as the appropriate primary basis for an introduction to the methods of algebraic topology. From homotopy theory, on the other hand, only the fundamental group and covering-space theory have traditionally been included among the basic initial concepts. Essentially all elementary classical textbooks of topology (the best of which is, in the opinion of the present authors, Seifert and Threlfalls A Textbook of Topology) begin with the homology theory of one or another classof complexes. Only at a later stage (and then still from a homological point of view) do fibre-space theory and the general problem of classifying homotopy classes of maps (homotopy theory) come in for consideration. However, methods developed in investigating the topology of differentiable manifolds, and intensively elaborated from the 1930s onwards (by Whitney and others), now permit a wholesale reorganization of the standard exposition Of the fundamentals of modern topology. In this new approach, which resembles more that of classical analysis, these fundamentals turn out to consist primarily of the elementary theory of smooth manifolds, homotopy theory based on these, and smooth fibre spaces. Furthermore, over the decade of the 1970s it became clear that exactly this complex of topological ideas and methods were proving to be fundamentally applicable in various areas of modern physics. 本書為英文版。

Preface CHAPTER 1 Homology and Cohomology. Computational Recipes 1.Cohomology groups as classes ofclosed differential forms Their homotopy invariance 2.The homology theory ofalgebraic 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 ofa product ofspaces. Multiplication in cohomology rings. The cohomology theory of H-spaces and Lie groups. The cohomology 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 opcration. Examples 10.2. Cohomology operations and Eilenberg-MacLane complexes 10.3. Computation of the rational homotopy groups 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 pheres 10.7. Stable homotopy classes ofmaps ofcell complexes 11. Homology theory and the fundamental group 12. The cohomology groups of hyperelliptic Riemann surfaces. Jacobitori. eodesics on multi-axis ellipsoids. Relationship to finite-gappotentials 13. The simplest properties of Kahler 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 ofsmooth 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 ofthe path space (m) 22. Applications of the index theorem 23. The periodic problem of the calculus of variations 24. Morse functions on 3-dimensioal 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 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 fonnula. 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) ofcombinatorial 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) Plateaus Problem. Spectral Bordisms and Globally Minimal Surfaces in Riemannian Manifolds Index Errata to Parts 1 and 11