Geometric Topology (eBook)

James C. Cantrell (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
712 Seiten
Elsevier Science (Verlag)
978-1-4832-7131-6 (ISBN)

Infos

Geometric Topology contains the proceedings of the 1977 Georgia Topology Conference, held at the University of Georgia on August 1977. The book is comprised of contributions from leading experts in the field of geometric topology.These contributions are grouped into four sections: low dimensional manifolds, topology of manifolds, shape theory and infinite dimensional topology, and miscellaneous problems. Subjects discussed under these sections include local spanning missing loops, the structure of generalized manifolds having nonmanifold set of trivial dimension, universal open principal fibrations, and how to build a flexible polyhedral surface. Topologists, geometers, and mathematicians will find the book very interesting and insightful.

Inhalt

Front Cover 1
Geometric Topology 4
Copyright Page 5
Table of Contents 8
Dedication 6
CONTRIBUTORS 12
PREFACE 14
PART I: LOW DIMENSIONAL MANIFOLDS 16
CHAPTER 1. A DECOMPOSITION OF WITHA NULL SEQUENCE OF CELLULAR ARCS 18
I. INTRODUCTION 19
II. THE TWO DISK PROPERTY 21
III. PRESERVING THE TWO DISK PROPERTY 23
IV. THE MAIN EXAMPLE 32
REFERENCES 36
CHAPTER 2. SPECIAL REPRESENTATIONS FOR 3-MANIFOLDS 38
I. INTRODUCTION 38
II. DEFINITIONS, NOTATION, CONVENTIONS 41
III. REPRESENTATIONS OF 3-MANIFOLDS BY SPECIAL FRAMED LINKS 46
IV. TIlE HEEGMRD DIAGRAMS AND SEWINGS ASSOCIATED TO CLASS 1, 2, 3 AND 4 FRAMED LINKS 54
V. QUESTIONS AND SPECULATIONS 63
REFERENCES 65
CHAPTER 3. SHRINKING COUNTABLE DECOMPOSITIONS OF E3 INTO POINTS AND TAME CELLS 68
I. INTRODUCTION 68
II. PUSHING A 1-COMPLEX OFF G* 70
III. PUSHING 2-CELLS OFF ELEMENTS OF G 75
IV. DECOMPOSITIONS OF E3 INTO POINTS' AND TAME 3-CELLS 79
V. COUNTABLE DECOMPOSITIONS OF E3 INTO POINTS AND TAME CELLS 86
REFERENCES 87
CHAPTER 4. LOCAL SPANNING MISSING LOOPS 88
REFERENCES 93
CHAPTER 5. A PROBLEM OF RUSHING 94
REFERENCES 101
CHAPTER 6. AN ADDITIVE INDEX THEOREM FOR CERTAINWILDLY EMBEDDED ARCS AND CURVES 102
REFERENCE 104
CHAPTER 7. SEIFERT FIBERED SPACES IN 3-MANIFOLDS 106
CHAPTER 8. ON EXOTIC HOMOTOPY EQUIVALENCES OF 3-MANIFOLDS 116
I. INTRODUCTION 116
II. EXOTIC HOMOTOPY EQUIVALENCES OF BALLS 119
III. INDICATION OF THE PROOF OF THE MAIN THEOREM 123
IV. APPENDIX 125
REFERENCES 126
CHAPTER 9. EIGHT FACES OF THE POINCARE HOMOLOGY 3-SPHERE 128
I. EIGHT DESCRIPTIONS 129
II. EQUIVALENCE OF THE DESCRIPTIONS 138
III. OTHER POSSIBILITIES 158
REFERENCES 160
CHAPTER 10. DECOMPOSING ANALYTIC SURFACES 162
I. DECOMPOSING 4-MANIFOLDS 162
II. ANALYTIC SURFACES 169
III. THE HYPERSURFACES OF CP3 183
IV. FAMILIES OF ANALYTIC MANIFOLDS AND THEIR DEGENERATIONS 192
V. IRRATIONAL CONNECTED SUMS 214
VI. ELLIPTIC SURFACES 218
VII. SURFACES OF GENERAL TYPE 225
VIII. AFTERWORD 229
REFERENCES 230
CHAPTER 11. HEEGAARD DIAGRAMS FOR CLOSED 4-MANIFOLDS 234
I. INTRODUCTION 234
II. THE HCl-fEOTOPY GROUP OF . # S1x S2 235
III. HEEGAARD SPLITTINGS 236
IV. HEEGAARD-DIAGRAMS FOR 4-MANIFOLDS 237
V. MOVES ON A HEEGAARD-DIAGRAM 238
VI. A PARTICULAR MODEL 240
VII. THE DUAL DIAGRAM 242
VIII. EXAMPLE: HEEGAARD DIAGRAMS FOR 2W4 243
IX. HEEGAARD DIAGIW4S FOR MAPPING TORUS 246
X. HEEGAARD DIAGRAMS FOR OPEN BOOKS WITH BINDING S2 250
REFERENCES 252
CHAPTER 12. DECOMPOSITIONS OF E3 WITH COUNTABLY MANY NON-DEGENERATE ELEMENTS 254
REFERENCES 267
CHAPTER 13. DECOMPOSITIONS OF E3 INTO CELLULAR SETS 268
REFERENCES 271
PART II: TOPOLOGY OF MANIFOLDS 274
CHAPTER 14. THE STRUCTURE OF GENERALIZED MANIFOLDS HAVING NONMANIFOLD SET OF TRIVIAL DIMENSION 276
I. INTRODUCTION 277
II. PREREQUISITES ON GENERALIZED MANIFOLDS 280
Ill. DETECTING CRUMPLED n-CELLS 281
IV• REPLACING CRUMPLED CELLS BY REAL CELLS 286
V. IMPROVING GENERALIZED MANIFOLDS 291
VI. RECOGNIZING MANIFOLDS 297
VII. SUMMARY 313
VIII. OPEN QUESTIONS 314
REFERENCES 314
CHAPTER 15. EMBEDDINGS AND IMMERSIONS OF FOUR DIMENSIONAL MANIFOLDS IN R6 316
REFERENCES 318
CHAPTER 16. GENERAL POSITION MAPS FOR.TOPOLOGICAL MANIFOLDS 320
APPLICATIONS 332
REFERENCES 335
CHAPTER 17. THE EQUIVARIANT SURGERY PROBLEM FOR I.NVOLUTIONS 338
REFERENCES 349
CHAPTER 18. THE DEGREE OF A BRANCHED COVERING OF A SPHERE 352
REFERENCES 358
CHAPTER 19. A UNIVERSAL 5-MANIFOLD WITH RESPECT TO SIMPLICIAL TRIANGULATIONS 360
I. INTRODUCTION 360
II. PRELIMINARY RESULTS 362
III. THE CONSTRUCTION 363
REFE.RENCES 365
CHAPTER 20. ON pi DIFF Mn 366
I. INTRODUCTION AND STATEMENT OF RESULTS 366
II. INDICATION OF THE PROOFS 370
REFERENCES 379
CHAPTER 21. CONSTRUCTION OF SURGERY PROBLEMS 382
I. INTRODUCTION 382
II. KEY HOMOTOPY PROPOSITIONS 383
III. BLOCKED NORMAL MAPS 390
IV. REDUCTION OF THEOREM A TO A SURGERY PROBLEM 394
V. COMPLETING SURGERY 396
APPENDIX 403
REFERENCES 405
CHAPTER 22 WILD EMBEDDINGS OF PIECEWISE LINEAR MANIFOLDS IN CODIMENSION TWO 408
I. INTRODUCTION 409
II. A BASIC CONSTRUCTION 410
III. A PL-SPINELESS MANIFOLD 417
IV. EXTENSION THEOREMS 419
V. WILD EMBEDDINGS OF S1 x S2n-1 INTO R 2n+2 424
VI. WILD EMBEDDINGS OF CLOSED SURFACES INTO R 4 427
VII. EXTENSION OF THE RESULTS 427
APPENDIX 433
REFERENCES 441
CHAPTER 23. FOLIATIONS AND MONOIDS OF EMBEDDINGS 444
I. INTRODUCTION 444
II. THE CONSTRUCTION OF fx. 448
III. THE IMMERSION-THEORETIC METHOD 454
REFERENCES 459
CHAPTER 24. ON THE TOPOLOGY OF NON-ISOLATED SINGULARITIES 460
I. INTRODUCTION 460
II. A REVIEW OF BASIC RESULTS 461
III. BASIC RESULTS FOR NON-ISOLATED SINGULARITI.ES 466
IV. X(F) AND p1(F) 471
V. THE HOMOLOGY OF F 475
VI. EXAMPLES 483
REFERENCES 486
CHAPTER 25. APPROXIMATING CAT CE MAPS BY CAT HOMEOMORPHISMS 490
I. INTRODUCTION 490
II. CAT CE MAPS 493
III. THE GROUP EH3 499
IV. THE PROOF OF THEOREM A 502
V. THE PROOF OF THEOREM B 506
VI. THE PROOF OF THEOREM C 513
REFERENCES 516
CHAPTER 26. ON COMPLEXES THAT ARE LIPSCHITZ MANIFOLDS 518
APPENDIX 0- LIPSCHITZ CONTINUATION 526
APPENDIX A. - LIPSCHITZ CONTINUATION TO SUSPENSION 530
APPENDIX B. - LIPSCHITZ ISOTOPY EXTENSIONS 534
REFERENCES 540
CHAPTER 27. ISOMETRIES OF INNER PRODUCT SPACESAND THEIR GEOMETRIC APPLICATIONS 542
I. INTRODUCTION: THE GEOMETRIC PROBLEMS 542
II. LOCALIZATION 547
III. COUPLING EXACT SEQUENCE 550
IV. OTHER EQUIVALENCE RELATIONS: METABOLIC, SPLIT AND HYPERBOLIC 551
REFERENCES 555
CHAPTER 28. HYPERBOLIC GEOMETRY AND HOMEOMORPHISMS 558
CONCLUDING REMARKS 565
HYPERBOLIC SPACE FORMS 568
CHAPTER 29. K2 AND DIFFEOMORPHISMS OF TWO AND THREE DIMENSIONAL MANIFOLDS 572
I. GENERAL PROPERTIES OF THE S INVARIANT 572
II. LOW DIMENSIONAL EXAMPLES 577
III. ANALYTIC TORSION 590
REFERENCES 591
PART III: SHAPE THEORY AND INFINITE DIMENSIONAL TOPOLOGY 594
CHAPTER 30. CONCORDANCES OF HILBERT CUBE MANIFOLDS AND TUBULAR NEIGHBORHOODS OF FINITE-DIMENSIONAL MANIFOLDS 596
I. INTRODUCTION 596
II. SOME LEMMAS FOR TIIEOREM 599
III. PROOF OF TIfEOREM 603
IV. SOME LEMMAS FOR THEOREM 2 604
V. PROOF OF THEOREM 2 608
REFERENCES 610
CHAPTER 31. UNIVERSAL OPEN PRINCIPAL FIBRATIONSI 612
I. INTRODUCTION 612
II. J. COHEN'S APPROXIMATION 613
III. CLASSIFICATION 614
IV. HOMOTOPY LIMITS 614
REFERENCES 615
CHAPTER 32. COMPACTA WITH THE HOMOTOPY TYPE OF FINITE COMPLEXES 618
REFERENCES 622
CHAPTER 33. MANIFOLDS MODELLED ON THE DIRECT LIMIT OF HILBERT CUBES 624
REFERENCES 634
CHAPTER 34. CONCORDANCE CLASSES OF FREE ACTIONS OF COMPACTLIE GROUPS ON INFINITE DIMENSIONIAL MANIFOLDS 636
REFERENCES 644
CHAPTER 35. CELL-LIKE MAPS, APPROXIMATE FIBRATIONS AND SHAPE FIBRATIONS 646
I. CELL-LIKE MAPS 646
II. APPROXIMATE FIBRATIONS 649
III. SHAPE FIBRATIONS 651
IV. CELL-LIKE MAPS WHICH ARE SHAPE FIBRATIONS 656
V. OBSTRUCTION THEORY FOR SHAPE FIBRATIONS WITH. APPLICATIONS TO CELL-LIKE MAPS 658
REFERENCES 663
CHAPTER 36. A WEAK FLATTENING CRITERION FOR COMPACTA IN 4-SPACE 664
I. INTRODUCTION 664
II. FINDING NEIGHBORHOODS WITH I-SPINES 666
IlI. PROOF OF THEOREM 1 669
REFERENCES 669
CHAPTER 37. BASED-FREE ACTIONS OF FINITE GROUPS ON HILBERT CUBES WITH ABSOLUTE RETRACT ORBIT SPACES ARE CONJUGATEI 670
I. INTRODUCTION 670
II. NOTATIONAL APPARATUS 673
III. LEMMAS 675
IV. PROOF OF THE MAIN THEOREM 679
V. CONCLUSION 684
VI. RELATED QUESTIONS AND PROBLEMS 685
REFERENCES 686
PART IV: MISCELLANEOUS PROBLEMS 688
CHAPTER 38. HOW TO BUILD A FLEXIBLE POLYHEDRAL SURFACE 690
I. INTRODUCTION 690
II. THE CONSTRUCTION 691
REFERENCES 698
CHAPTER 39. A SHORT LIST OF PROBLEMS 700
REFERENCE 705
CHAPTER 40. SOME BORSUK-ULAM TYPE THEOREMS 708
REFERENCES 712

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-7131-5 / 1483271315
• ISBN-13: 978-1-4832-7131-6 / 9781483271316