Circle-valued morse theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
de Gruyter
2006
|
| Schriftenreihe: | De Gruyter studies in mathematics
32 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | IX, 454 S. graph. Darst. |
| ISBN: | 3110158078 9783110158076 |
Internformat
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Datensatz im Suchindex
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ANDREI V. PAJITNOV CIRCLE-VALUED MORSE THEORY W DE G WALTER DE GRUYTER *
BERLIN * NEW YORK CONTENTS PREFACE 1 INTRODUCTION 5 PART 1. MORSE
FUNCTIONS AND VECTOR FIELDS ON MANIFOLDS . . 15 CHAPTER 1. VECTOR FIELDS
AND C TOPOLOGY 17 1. MANIFOLDS WITHOUT BOUNDARY 17 2. COBORDISMS 24
CHAPTER 2. MORSE FUNCTIONS AND THEIR GRADIENTS 33 1. MORSE FUNCTIONS AND
MORSE FORMS 35 2. GRADIENTS OF MORSE FUNCTIONS AND FORMS 50 3. MORSE
FUNCTIONS ON COBORDISMS 62 CHAPTER 3. GRADIENT FLOWS OF REAL-VALUED
MORSE FUNCTIONS 67 1. LOCAL PROPERTIES OF GRADIENT FLOWS 67 2.
DESCENDING DISCS 81 3. THE GRADIENT DESCENT 99 EXERCISES TO PART 1 108
PART 2. TRANSVERSALITY, HANDLES, MORSE COMPLEXES 109 CHAPTER 4. THE
KUPKA-SMALE TRANSVERSALITY THEORY FOR GRADIENT FLOWS ILL 1. PERTURBING
THE LYAPUNOV DISCS 112 2. THE TRANSVERSE GRADIENTS ARE GENERIC 122 3.
ALMOST TRANSVERSE GRADIENTS AND REARRANGEMENT LEMMA . . . 132 CHAPTER 5.
HANDLES 163 1. CONSTRUCTION OF HANDLES 163 2. MORSE FUNCTIONS AND THE
CELLULAR STRUCTURE OF MANIFOLDS . . . . 171 VIII CONTENTS 3. HOMOLOGY OF
ELEMENTARY COBORDISMS 172 4. APPENDIX: ORIENTATIONS, COORIENTATIONS,
FUNDAMENTAL CLASSES ETC 189 CHAPTER 6. THE MORSE COMPLEX OF A MORSE
FUNCTION 195 1. THE MORSE COMPLEX FOR TRANSVERSE GRADIENTS 196 2. THE
MORSE COMPLEX FOR ALMOST TRANSVERSE GRADIENTS 208 3. THE MORSE CHAIN
EQUIVALENCE 212 4. MORE ABOUT THE MORSE COMPLEX 218 HISTORY AND SOURCES
227 PART 3. CELLULAR GRADIENTS 229 CHAPTER 7. CONDITION ( ) 231 1. THE
GRADIENT DESCENT REVISITED 232 2. DEFINITION AND FIRST PROPERTIES OF
CELLULAR GRADIENTS 238 CHAPTER 8. CELLULAR GRADIENTS ARE C-GENERIC 243
1. INTRODUCTION 243 2. THE STRATIFIED GRADIENT DESCENT 244 3. QUICK
FLOWS 254 4. PROOF OF THE ^-APPROXIMATION THEOREM 274 CHAPTER 9.
PROPERTIES OF CELLULAR GRADIENTS 281 1. HOMOLOGICAL GRADIENT DESCENT 282
2. CYCLIC COBORDISMS AND ITERATIONS OF THE GRADIENT DESCENT MAP 287 3.
HANDLE-LIKE NITRATIONS ASSOCIATED WITH THE CELLULAR GRADIENTS . 301
SOURCES . 321 PART 4. CIRCLE-VALUED MORSE MAPS AND NOVIKOV COMPLEXES 323
CHAPTER 10. COMPLETIONS OF RINGS, MODULES AND COMPLEXES 325 1. CHAIN
COMPLEXES OVER A[[T]] 325 2. NOVIKOV RINGS 332 CHAPTER 11. THE NOVIKOV
COMPLEX OF A CIRCLE-VALUED MORSE MAP . . 335 1. THE NOVIKOV COMPLEX FOR
TRANSVERSE GRADIENTS 336 2. NOVIKOV HOMOLOGY 340 3. THE NOVIKOV COMPLEX
FOR ALMOST TRANSVERSE GRADIENTS 346 4. EQUIVARIANT MORSE EQUIVALENCES
348 5. ON THE SINGULAR CHAIN COMPLEX OF THE INFINITE CYCLIC COVERING .
354 CONTENTS IX 6. THE CANONICAL CHAIN EQUIVALENCE 356 7. MORE ABOUT THE
NOVIKOV COMPLEX 359 CHAPTER 12. CELLULAR GRADIENTS OF CIRCLE-VALUED
MORSE FUNCTIONS AND THE RATIONALITY THEOREM 367 1. NOVIKOV'S EXPONENTIAL
GROWTH CONJECTURE 369 2. THE BOUNDARY OPERATORS IN THE NOVIKOV COMPLEX
371 3. CELLULAR GRADIENTS OF CIRCLE-VALUED MORSE FUNCTIONS 375 4.
GRADIENT-LIKE VECTOR FIELDS AND RIEMANNIAN GRADIENTS 380 CHAPTER 13.
COUNTING CLOSED ORBITS OF THE GRADIENT FLOW 383 1. LEFSCHETZ ZETA
FUNCTIONS OF THE GRADIENT FLOWS 383 2. HOMOLOGICAL AND DYNAMICAL
PROPERTIES OF THE CELLULAR GRADIENTS 387 3. WHITEHEAD GROUPS AND
WHITEHEAD TORSION 399 4. THE WHITEHEAD TORSION OF THE CANONICAL CHAIN
EQUIVALENCE . . 407 CHAPTER 14. SELECTED TOPICS IN THE MORSE-NOVIKOV
THEORY 413 1. HOMOLOGY WITH LOCAL COEFFICIENTS AND THE DE RHAM FRAMEWORK
FOR THE MORSE-NOVIKOV THEORY 413 2. THE UNIVERSAL NOVIKOV COMPLEX 417 3.
THE MORSE-NOVIKOV THEORY AND FIBRING OBSTRUCTIONS 419 4. EXACTNESS
THEOREMS AND LOCALIZATION CONSTRUCTIONS 421 5. THE MORSE-NOVIKOV THEORY
OF CLOSED 1-FORMS 422 6. CIRCLE-VALUED MORSE THEORY FOR KNOTS AND LINKS
424 HISTORY AND SOURCES 435 BIBLIOGRAPHY 437 SELECTED SYMBOLS AND
ABBREVIATIONS 445 SUBJECT INDEX 449 |
| adam_txt |
ANDREI V. PAJITNOV CIRCLE-VALUED MORSE THEORY W DE G WALTER DE GRUYTER *
BERLIN * NEW YORK CONTENTS PREFACE 1 INTRODUCTION 5 PART 1. MORSE
FUNCTIONS AND VECTOR FIELDS ON MANIFOLDS . . 15 CHAPTER 1. VECTOR FIELDS
AND C TOPOLOGY 17 1. MANIFOLDS WITHOUT BOUNDARY 17 2. COBORDISMS 24
CHAPTER 2. MORSE FUNCTIONS AND THEIR GRADIENTS 33 1. MORSE FUNCTIONS AND
MORSE FORMS 35 2. GRADIENTS OF MORSE FUNCTIONS AND FORMS 50 3. MORSE
FUNCTIONS ON COBORDISMS 62 CHAPTER 3. GRADIENT FLOWS OF REAL-VALUED
MORSE FUNCTIONS 67 1. LOCAL PROPERTIES OF GRADIENT FLOWS 67 2.
DESCENDING DISCS 81 3. THE GRADIENT DESCENT 99 EXERCISES TO PART 1 108
PART 2. TRANSVERSALITY, HANDLES, MORSE COMPLEXES 109 CHAPTER 4. THE
KUPKA-SMALE TRANSVERSALITY THEORY FOR GRADIENT FLOWS ILL 1. PERTURBING
THE LYAPUNOV DISCS 112 2. THE TRANSVERSE GRADIENTS ARE GENERIC 122 3.
ALMOST TRANSVERSE GRADIENTS AND REARRANGEMENT LEMMA . . . 132 CHAPTER 5.
HANDLES 163 1. CONSTRUCTION OF HANDLES 163 2. MORSE FUNCTIONS AND THE
CELLULAR STRUCTURE OF MANIFOLDS . . . . 171 VIII CONTENTS 3. HOMOLOGY OF
ELEMENTARY COBORDISMS 172 4. APPENDIX: ORIENTATIONS, COORIENTATIONS,
FUNDAMENTAL CLASSES ETC 189 CHAPTER 6. THE MORSE COMPLEX OF A MORSE
FUNCTION 195 1. THE MORSE COMPLEX FOR TRANSVERSE GRADIENTS 196 2. THE
MORSE COMPLEX FOR ALMOST TRANSVERSE GRADIENTS 208 3. THE MORSE CHAIN
EQUIVALENCE 212 4. MORE ABOUT THE MORSE COMPLEX 218 HISTORY AND SOURCES
227 PART 3. CELLULAR GRADIENTS 229 CHAPTER 7. CONDITION ( ) 231 1. THE
GRADIENT DESCENT REVISITED 232 2. DEFINITION AND FIRST PROPERTIES OF
CELLULAR GRADIENTS 238 CHAPTER 8. CELLULAR GRADIENTS ARE C-GENERIC 243
1. INTRODUCTION 243 2. THE STRATIFIED GRADIENT DESCENT 244 3. QUICK
FLOWS 254 4. PROOF OF THE ^-APPROXIMATION THEOREM 274 CHAPTER 9.
PROPERTIES OF CELLULAR GRADIENTS 281 1. HOMOLOGICAL GRADIENT DESCENT 282
2. CYCLIC COBORDISMS AND ITERATIONS OF THE GRADIENT DESCENT MAP 287 3.
HANDLE-LIKE NITRATIONS ASSOCIATED WITH THE CELLULAR GRADIENTS . 301
SOURCES . 321 PART 4. CIRCLE-VALUED MORSE MAPS AND NOVIKOV COMPLEXES 323
CHAPTER 10. COMPLETIONS OF RINGS, MODULES AND COMPLEXES 325 1. CHAIN
COMPLEXES OVER A[[T]] 325 2. NOVIKOV RINGS 332 CHAPTER 11. THE NOVIKOV
COMPLEX OF A CIRCLE-VALUED MORSE MAP . . 335 1. THE NOVIKOV COMPLEX FOR
TRANSVERSE GRADIENTS 336 2. NOVIKOV HOMOLOGY 340 3. THE NOVIKOV COMPLEX
FOR ALMOST TRANSVERSE GRADIENTS 346 4. EQUIVARIANT MORSE EQUIVALENCES
348 5. ON THE SINGULAR CHAIN COMPLEX OF THE INFINITE CYCLIC COVERING .
354 CONTENTS IX 6. THE CANONICAL CHAIN EQUIVALENCE 356 7. MORE ABOUT THE
NOVIKOV COMPLEX 359 CHAPTER 12. CELLULAR GRADIENTS OF CIRCLE-VALUED
MORSE FUNCTIONS AND THE RATIONALITY THEOREM 367 1. NOVIKOV'S EXPONENTIAL
GROWTH CONJECTURE 369 2. THE BOUNDARY OPERATORS IN THE NOVIKOV COMPLEX
371 3. CELLULAR GRADIENTS OF CIRCLE-VALUED MORSE FUNCTIONS 375 4.
GRADIENT-LIKE VECTOR FIELDS AND RIEMANNIAN GRADIENTS 380 CHAPTER 13.
COUNTING CLOSED ORBITS OF THE GRADIENT FLOW 383 1. LEFSCHETZ ZETA
FUNCTIONS OF THE GRADIENT FLOWS 383 2. HOMOLOGICAL AND DYNAMICAL
PROPERTIES OF THE CELLULAR GRADIENTS 387 3. WHITEHEAD GROUPS AND
WHITEHEAD TORSION 399 4. THE WHITEHEAD TORSION OF THE CANONICAL CHAIN
EQUIVALENCE . . 407 CHAPTER 14. SELECTED TOPICS IN THE MORSE-NOVIKOV
THEORY 413 1. HOMOLOGY WITH LOCAL COEFFICIENTS AND THE DE RHAM FRAMEWORK
FOR THE MORSE-NOVIKOV THEORY 413 2. THE UNIVERSAL NOVIKOV COMPLEX 417 3.
THE MORSE-NOVIKOV THEORY AND FIBRING OBSTRUCTIONS 419 4. EXACTNESS
THEOREMS AND LOCALIZATION CONSTRUCTIONS 421 5. THE MORSE-NOVIKOV THEORY
OF CLOSED 1-FORMS 422 6. CIRCLE-VALUED MORSE THEORY FOR KNOTS AND LINKS
424 HISTORY AND SOURCES 435 BIBLIOGRAPHY 437 SELECTED SYMBOLS AND
ABBREVIATIONS 445 SUBJECT INDEX 449 |
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| language | English |
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| publisher | de Gruyter |
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| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Pajitnov, Andrei V. Verfasser aut Circle-valued morse theory Andrei V. Pajitnov Berlin [u.a.] de Gruyter 2006 IX, 454 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 32 Manifolds (Mathematics) Morse theory Morse-Theorie (DE-588)4197103-6 gnd rswk-swf Morse-Theorie (DE-588)4197103-6 s DE-604 De Gruyter studies in mathematics 32 (DE-604)BV000005407 32 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2848374&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015822104&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pajitnov, Andrei V. Circle-valued morse theory De Gruyter studies in mathematics Manifolds (Mathematics) Morse theory Morse-Theorie (DE-588)4197103-6 gnd |
| subject_GND | (DE-588)4197103-6 |
| title | Circle-valued morse theory |
| title_auth | Circle-valued morse theory |
| title_exact_search | Circle-valued morse theory |
| title_exact_search_txtP | Circle-valued morse theory |
| title_full | Circle-valued morse theory Andrei V. Pajitnov |
| title_fullStr | Circle-valued morse theory Andrei V. Pajitnov |
| title_full_unstemmed | Circle-valued morse theory Andrei V. Pajitnov |
| title_short | Circle-valued morse theory |
| title_sort | circle valued morse theory |
| topic | Manifolds (Mathematics) Morse theory Morse-Theorie (DE-588)4197103-6 gnd |
| topic_facet | Manifolds (Mathematics) Morse theory Morse-Theorie |
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