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Mathematics Series)

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Author B.A.Dubrovin, A.T.Fomenko, S.P.Novikov

Publisher World Publishing Corporation

Isbn 7506212641, 9787506212649

Published Nov 1999

Pages 416

Size 24k

Edition first

Format Paperback

Language English

Page material copper plate paper

Shop price: $23.80

Product Description:

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 Threlfall's 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.

Content:

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

Index

Errata to Parts 1 and 11

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