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Modern geometry: methods and applications 3 Introduction to homology theory

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Bibliographische Detailangaben
Beteiligte Personen:	Dubrovin, Boris Anatol'evič 1950-2019 (VerfasserIn), Fomenko, Anatolij Timofeevič 1945- (VerfasserIn), Novikov, Sergej P. 1938-2024 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York u.a. Springer 1990
Schriftenreihe:	Graduate texts in mathematics 124
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002631101&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	IX, 416 S.
ISBN:	0387972714 3540972714

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Datensatz im Suchindex

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adam_text	Contents Preface v CHAPTER 1 Homology and Cohomology. Computational Recipes 1 §1. Cohomology groups as classes of closed differential forms. Their homotopy invariance 1 §2. The homology theory of algebraic complexes 17 §3. Simplicial complexes. Their homology and cohomology groups. The classification of the two-dimensional closed surfaces 24 §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 42 §5. The singular homology and cohomology groups. Their homotogy invariance. The exact sequence of a pair. Relative homology groups 59 §6. The singular homology of cell complexes. Its equivalence with cell homology. Poincaré duality in simplicial homology 72 §7. The homology groups of a product of spaces. Multiplication in cohomology rings. The cohomology theory of tf-spaces and Lie groups. The cohomology of the unitary groups 83 §8. The homology theory of fibre bundles (skew products) 97 §9. The extension problem for maps, homotopies, and cross-sections. Obstruction cohomology classes 1 Ю 9.1. The extension problem for maps ПО 9.2. The extension problem for homotopies 112 9.3. The extension problem for cross-sections 113 §10. Homology theory and methods for computing homotopy groups. The Cartan-Serre theorem. Cohomology operations. Vector bundles. I Î8 10.1. The concept of a cohomology operation. Exampies 118 10.2. Cohomology operations and Eilenberg-MacLane complexes 122 viii Contents 10.3. Computation of the rational homotopy groups π, ® Q 126 10.4. Application to vector bundles. Characteristic classes 129 10.5. Classification of the Steenrod operations in low dimensions 135 10.6. Computation of the first few nontrivial stable homotopy groups of spheres 143 10.7. Stable homotopy classes of maps of cell complexes 149 §11. Homology theory and the fundamental group 152 §12. The cohomology groups of hyperelliptic Riemann surfaces. Jacobi tori. Geodesies on multi-axis ellipsoids. Relationship to finite-gap potentials 160 §13. The simplest properties of Kahler manifolds. Abelian tori 174 §14. Sheaf cohomology 180 CHAPTER 2 Critical Points of Smooth Functions and Homology Theory 188 §15. Morse functions and cell complexes 188 §16. The Morse inequalities 195 §17. Morse-Smale functions. Handles. Surfaces 202 §18. Poincaré duality 216 §19. Critical points of smooth functions and the Lyusternik- Shnirelman category of a manifold 223 §20. Critical manifolds and the Morse inequalities. Functions with symmetry 238 §21. Critical points of functionals and the topology of the path space Ω(Λί) 247 §22. Applications of the index theorem 262 §23. The periodic problem of the calculus of variations 270 §24. Morse functions on 3-dimensional manifolds and Heegaard splittings 277 §25. Unitary Bott periodicity and higher-dimensional variational problems 283 25.1. The theorem on unitary periodicity 283 25.2. Unitary periodicity via the two-dimensional calculus of variations 293 25.3. Onthogonal periodicity via the higher-dimensional calculus of variations 302 §26. Morse theory and certain motions in the planar η -body problem 310 CHAPTER 3 Cobordisms and Smooth Structures 325 §27. Characteristic numbers. Cobordisms. Cycles and submanifolds. The signature of a manifold 325 27.1. Statement of the problem. The simplest facts about cobordisms. The signature 325 27.2. Thorn complexes. Calculation of cobordisms (modulo torsion). The signature formula. Realization of cycles as submanifolds 334 27.3. Some applications of the signature formula. The signature and the problem of the invariance of classes 351 Contents ix §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 357 Bibliography 372 APPENDIX 1 (by S. P. Novikov) An Analogue of Morse Theory for Many-Valued Functions. Certain Properties of Poisson Brackets 377 APPENDIX 2 (by A. T. Fomenko) Plateau s Problem. Spectral Bordisms and Globally Minimal Surfaces in Riemannian Manifolds 391 Index 407 Errata to Parts I and II 413
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author	Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024
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isbn	0387972714 3540972714
language	English
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spellingShingle	Dubrovin, Boris Anatol'evič 1950-2019 Fomenko, Anatolij Timofeevič 1945- Novikov, Sergej P. 1938-2024 Modern geometry methods and applications Graduate texts in mathematics
title	Modern geometry methods and applications
title_alt	Sovremennaja geometrija
title_auth	Modern geometry methods and applications
title_exact_search	Modern geometry methods and applications
title_full	Modern geometry methods and applications 3 Introduction to homology theory B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov
title_fullStr	Modern geometry methods and applications 3 Introduction to homology theory B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov
title_full_unstemmed	Modern geometry methods and applications 3 Introduction to homology theory B. A. Dubrovin ; A. T. Fomenko ; S. P. Novikov
title_short	Modern geometry
title_sort	modern geometry methods and applications introduction to homology theory
title_sub	methods and applications
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002631101&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV005835289 (DE-604)BV000000067
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