Fete of Topology (eBook)

Papers Dedicated to Itiro Tamura

Y. Matsumoto, T. Mizutani, S. Morita (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
614 Seiten
Elsevier Science (Verlag)
978-1-4832-5918-5 (ISBN)

Infos

A Fête of Topology: Papers Dedicated to Itiro Tamura focuses on the progress in the processes, methodologies, and approaches involved in topology, including foliations, cohomology, and surface bundles. The publication first takes a look at leaf closures in Riemannian foliations and differentiable singular cohomology for foliations. Discussions focus on differentiable singular chains restricted to leaves, differentiable singular cohomology for foliations, covering of pseudogroups and fundamental group, normal type of an orbit closure, and construction of a global model. The text then takes a look at measure of exceptional minimal sets of codimension one foliations, examples of exceptional minimal sets, foliations transverse to non-singular Morse-Smale flows, and Chern character for discrete groups. The manuscript ponders on characteristic classes of surface bundles and bounded cohomology, Hill's equation, isomonodromy deformation and characteristic classes, and topology of folds, cusps, and Morin singularities. Topics include system of Hill's equations, Lagrange-Grassman manifold, positive curves, Morse theory, bounded cohomology, and characteristic classes of surface bundles. The publication is a vital source of information for researchers interested in topology.

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A Fete of Topology: Papers Dedicated to Itiro Tamura focuses on the progress in the processes, methodologies, and approaches involved in topology, including foliations, cohomology, and surface bundles. The publication first takes a look at leaf closures in Riemannian foliations and differentiable singular cohomology for foliations. Discussions focus on differentiable singular chains restricted to leaves, differentiable singular cohomology for foliations, covering of pseudogroups and fundamental group, normal type of an orbit closure, and construction of a global model. The text then takes a look at measure of exceptional minimal sets of codimension one foliations, examples of exceptional minimal sets, foliations transverse to non-singular Morse-Smale flows, and Chern character for discrete groups. The manuscript ponders on characteristic classes of surface bundles and bounded cohomology, Hill's equation, isomonodromy deformation and characteristic classes, and topology of folds, cusps, and Morin singularities. Topics include system of Hill's equations, Lagrange-Grassman manifold, positive curves, Morse theory, bounded cohomology, and characteristic classes of surface bundles. The publication is a vital source of information for researchers interested in topology.

Inhalt

Front Cover 1
A Fête of Topology Papers Dedicated to Itiro Tamura 4
Copyright Page 5
Table of Contents 6
Contributors 8
Foreword 10
Publications of Itiro Tamura 12
PART 1: FOLIATIONS 14
CHAPTER 1. LEAF CLOSURES IN RIEMANNIAN FOLIATIONS 16
1. Complete pseudogroups of isometries 17
2. Covering of pseudogroups and fundamental group 22
3. Closure of a complete pseudogroup 26
4. Normal type of an orbit closure 29
5. Construction of a global model 34
6. Applications to Riemannian foliations 40
References 45
CHAPTER 2. ON THE HOMOMORPHISM H* (BIRd) . H*BDiff 8c(IR)d) 46
1 Introduction 46
2 Tori in BGd 46
3 Homoloq y class {F(a)} 48
4 Reductio of the theorem 51
5 Prolongation 52
6 Completion of the proof 56
References 60
CHAPTER 3. THE GODBILLON-VEY COCYCLE OF DiffRn 62
Introduction 62
1 The double complex A p,q 63
2 The cocycle in An+1,n 65
3 The cocycle in C2n+1(F) 68
4 Some remarks 73
References 74
CHAPTER 4. DIFFERENTIABLE SINGULAR COHOMOLOGY FOR FOLIATIONS 76
Introduction 76
1 Differentiable singular chains restricted to leaves 77
2 Differentiable cochains for C*(M R)
3 Differentiable singular cohomology for foliations 86
References 92
CHAPTER 5. MEASURE OF EXCEPTIONAL MINIMAL SETS OF CODIMENSION ONE FOLIATIONS 94
1 . Introduction 94
2. Basic inequalities 96
3. Preparations in symbolic dynamics 99
4 . Proof of Theorem 1 103
REFERENCES 107
CHAPTER 6. EXAMPLES OF EXCEPTIONAL MINIMAL SETS 108
1. Exceptional minimal sets in piecewise linear foliations 108
2. Branched staircases 110
References 112
CHAPTER 7. AVERAGE SIGNATURES OF PA-LEAVES OF CODIMENSION-ONE FOLIATIONS 114
1. Introduction 114
2. Average signatures 116
3. Examples 118
4. A qualitative theory of PA leaves 122
5. The proof of Main Theorem 127
REFERENCES 138
CHAPTER 8. FOLIATIONS TRANSVERSE TO NON-SINGULAR MORSE-SMALE FLOWS 140
1. Preliminaries and known results 142
2. Preparation 144
3. Application 160
References 171
PART 2: CHARACTERISTIC CLASSES 174
CHAPTER 9. CHERN CHARACTER FOR DISCRETE GROUPS 176
1. G finite: Chern character 180
2. G finite: Sheaf theory 184
3. G finite: Homotopy quotient 188
4. G finite: Integration over the fiber 191
5. G finite: .a(t*,G) 193
6. G finite: Index Theorem 195
7. G countable: Chern character for proper actions 198
8. Proper actions: Integration over the fiber 205
9. Proper actions: Td(t* ,G) 207
11. Twisted homology and K homology 212
12. Improper actions: Integration over the fiber 215
13. Improper actions: Td(t ,G) 219
15. Cher n character for improper actions 228
Appendix 1: CO(X) x G. Proof of Lemma (7.5) 232
Appendix 2: G uncountable 237
Appendix 3: Classifying space for proper actions 240
References 243
CHAPTER 10. CHARACTERISTIC CLASSES OF SURFACE BUNDLES AND BOUNDED COHOMOLOGY 246
1. Introduction 246
2. Characteristic classes of surface bundles 246
3. Bounded cohomology 248
4. The Euler class for flat S1-bundles 250
5. The Euler class for flat PSL2R-bundles 255
6. A theorem 262
7. Surface bundles with amenable holonomy 266
References 268
CHAPTER 11. HILL'S EQUATION, ISOMONODROMY DEFORMATION AND CHARACTERISTIC CLASSES 272
1. Introduction 272
2. System of Hill's equations 275
3. Isomonodromy deformation 277
4. Lagrange-Grassmann manifold 279
5. Positive curves 281
6. Morse theory 283
7. Homotopy type of space of positive loops 290
8. Pontrjaqin classes 296
9. Morse index 298
10. Appendix: Isomonodromy in Lagrange-Grassmann 302
References 303
PART 3: SINGULARITIES AND ORBIFOLDS 304
CHAPTER 12. POLYEDRES EVANESCENTS ET EFFONDREMENTS 306
1 Introduction 306
1. RAPPELS 308
2. CAS DE LA DIMENSION DEUX 315
3. LA CONSTRUCTION DU SQUELETTE 325
4. LE THEOREME PRINCIPAL 330
5. DEMONSTRATION DE LA RECURRENCE 332
Références 341
CHAPTER 13. TOPOLOGY OF FOLDS , CUSPS AND MORIN SINGULARITIES 344
Introduction 344
2. Lemmas for Theorem 1 346
3. Lemmas for Theorem 2 348
4. Lemmas for Theorem 3 350
5. Proof of Theorem 1 351
6. Proof of Theorem 2 352
7. Proof of Theorem 3 353
8. Proof of Lemma 2.3 354
9. Proof of Lemma 2.4 357
10. Proof of Lemma 3.2 358
11 . Proof of Lemma 4.2 362
Bibliography 366
CHAPTER 14. EXAMPLES OF ALGEBRAIC SURFACES WITH q = 0 AND p = 1 WHICH ARE LOCALLY HYPERSURFACSE 368
1 Introduction 368
2 Compactification 369
3 Surfaces with q = pg= 0 371
4 Surfaces with q = 0 and p g= 1 373
References 375
CHAPTER 15. ON BIHOLOMORPHISMS BETWEEN SOME KUMMER BRANCHED COVERING SPACES OF COMPLEX PROJECTIVE PLANE 378
1. Introduction 378
2. Complex reflection groups of normal complex analytic surfaces 379
3. Kummer branched coverings as complex reflection groups 382
4. The automorphism groups of Kummer branched covering curves 387
5. Equivariant resolution of K( P2:nL ) and the proof of Theorem 392
References 402
CHAPTER 16. A COMPACTNESS THEOREM OF A SET OF ASPHERICAL RIEMANNIAN ORBIFOLDS 404
0 Introduction 404
1 Aspherical orbifolds with small volume 407
2 Proofs of Theorems 0-9 , 0-10 and 0-11 408
3 Preliminaries to the proof of Theorem 0-6 410
4 The limit of the universal covering spaces 415
5 The action of the group G 420
References 425
PART 4: 3 AND 4-DIMENSIONAL MANIFOLDS 428
CHAPTER 17. VIRTUAL BETTI NUMBERS OF SOME HYPERBOLIC 3-MANIFOLDS 430
1 Introduction 430
2 Computation I 431
3 (3,2q ) Turk's head knots 433
4 Lifting R and D 440
5 Computation II 443
6 Virtual Betti numbers 445
References 450
CHAPTER 18. QUASI-LOCALNESS AND UNKNOTTING THEOREMS FOR KNOTS IN 3-MANIFOLDS 452
References 455
CHAPTER 19. ENERGY OF A KNOT 456
1 Introduction 456
2 Spaces of knots and loops 457
3 Energy of a knot 460
4 How to reduce energy of a knot 460
CHAPTER 20. KNOTS I N THE STABLE 4-SPACE AN OVERVIEW
1 The stable 4-space 467
2 The groups of links 469
3 The torsion pairing invariant 473
4 Knots with no minimal Seifert manifolds and the supporting degree 475
5 Cobordism 477
6 Arithematic 478
References 482
CHAPTER 21. ON THE 4-DIMENSIONAL SEIFERT FIBERINGS WITH EUCLIDEAN BASE ORBIFOLDS 484
0 Preliminaries 485
1 The cases with orientable base orbifolds 488
2 T2 -bundles over the Klein bottle 495
3 The Seifert fiberings with base P2 (2,2) 509
4. Main Theorem 529
References 535
CHAPTER 22. ON IMMERSED 2-SPHERES IN S2X S2 538
1 Introduction 538
2 Main Lemma and Proofs of Theorems 541
3 Proof of Main Lemma 545
References 554
CHAPTER 23. AN EXPLICIT FORMULA OF THE METRIC ON THE MODULI SPACE OF BPST-INSTANTONS 556
0. Introduction 556
1. Generic BPST-instantons 557
2. Parametrization of M 559
3. Computation of the metric 561
4. Sectional curvature 565
References 569
CHAPTER 24. HYPERBOLIC METRIC ON THE MODULI SPACE OF BPST-INSTANTONS OVER S 570
References 574
PART 5: GROUP ACTIONS 576
CHAPTER 25. An s-COBORDISM THEOREM FOR SEMI-FREE S1-MANIFOLDS 578
0 Introduction 578
1 Excision of deformation retractions of orbit spaces 580
2 Blowing up along the fixed point set of codimension two 585
3 Heredity of vanishing of equivariant Whitehead torsions 587
References 595
CHAPTER 26. SMOOTH GROUP ACTIONS ON COHOMOLOGY COMPLEX PROJECTIVE SPACES WITH A FIXED POINT COMPONENT OF CODIMENSION 2 598
0 Introduction 598
1 Statement of results 599
2 Equivariant Gysin homomorphism 603
3 Proof of Theorem A 607
4 Proof of Theorem C 608
5 Examples 611
References 613

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-5918-8 / 1483259188
• ISBN-13: 978-1-4832-5918-5 / 9781483259185