An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2018]
|
| Series: | Memoirs of the American Mathematical Society
volume 256, number 1226 (second of 6 numbers) |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 234, 4 ungezählte Seiten |
| ISBN: | 9781470414214 |
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MARC
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| adam_text | Contents
Preface xiii
Acknowledgments xiv
Chapter 1. Introduction 1
1.1. Summary of main results 1
1.2. Outline of the argument 4
1.2.1. Problem of overlaps 5
1.2.2. Overlap space and overlap maps 6
1.2.3. Associativity of splicing maps 7
1.2.4. Inst ant on moduli space with spliced ends 8
1.2.5. Space of global splicing data 9
1.2.6. Definition of link of a subspace of a moduli space of ideal
Seiberg-Witten monopoles 9
1.2.7. Computation of intersection numbers with the link of the moduli
space of ideal Seiberg-Witten monopoles 10
1.3. Kotschick-Morgan Conjecture 10
1.4. Outline of the monograph 12
Chapter 2. Preliminaries 13
2.1. The moduli space of SO(3) monopoles 13
2.1.1. Clifford modules 13
2.1.2. SO(3) monopoles 14
2.2. Stratum of anti-self-dual or zero-section solutions 16
2.3. Strata of Seiberg-Witten or reducible solutions 16
2.3.1. Seiberg-Witten monopoles 17
2.3.2. Seiberg-Witten invariants 17
2.3.3. Reducible SO(3) monopoles 18
2.3.4. Circle actions 19
2.3.5. The virtual normal bundle of the Seiberg-Witten moduli space 19
2.4. Cohomology classes on the moduli space of SO(3) monopoles 21
2.5. Donaldson invariants 22
2.6. Links and the cobordism 23
Chapter 3. Diagonals of symmetric products of manifolds 25
3.1. Definitions 25
3.1.1. Subgroups of the symmetric group 25
3.1.2. Definition of the diagonals 26
3.1.3. Strata of the symmetric product 27
3.2. Incidence relations among diagonals and strata 27
3.3. Normal bundles of diagonals and strata 28
V
VI
CONTENTS
3.4. Enumeration of the strata 30
Chapter 4. A partial Thom-Mather structure on symmetric products 31
4.1. Introduction 31
4.2. Diagonals in products of M4 32
4.3. Families of metrics 35
4.4. Overlap maps 37
4.4.1. The downwards overlap map 38
4.4.2. The upwards overlap map 38
4.4.3. Commuting overlap maps 39
4.4.4. The projection maps 40
4.5. Construction of the families of locally flattened metrics 40
4.6. Normal bundles of strata of Sym^(JA) 42
4.7. The tubular distance function 44
4.8. Decomposition of the strata 47
Chapter 5. The instanton moduli space with spliced ends 51
5.1. Introduction 51
5.2. Connections over the four-dimensional sphere 53
5.3. Strata containing the product connection 54
5.3.1. Tubular neighborhoods 55
5.4. The splicing map with the product connection over M4 57
5.5. Composition of splicing maps 60
5.5.1. Definition of the overlap data 61
5.5.2. Equality of splicing maps 64
5.5.3. Symmetric group actions and quotients 66
5.6. The spliced end of the instanton moduli space 67
5.7. Tubular neighborhoods of the instanton moduli space with spliced
ends 74
5.8. Isotopy of the spliced end of the instanton moduli space 76
5.9. Properties of the instanton moduli space with spliced ends 78
Chapter 6. The space of global splicing data 81
6.1. Introduction 81
6.2. Splicing data 82
6.2.1. Background pairs 82
6.2.2. Riemannian metrics 82
6.2.3. Frame bundles 82
6.2.4. Group actions on the frame bundles 83
6.2.5. Space of splicing data 83
6.3. The flattening map on pairs 84
6.4. The crude splicing map 88
6.4.1. The standard splicing map 89
6.4.2. Construction of the crude splicing map 90
6.4.3. Properties of the crude splicing map 91
6.5. Overlap spaces and maps 92
6.5.1. The overlap space 93
6.5.2. The upwards overlap map 94
6.5.3. Downwards overlap map 95
CONTEXTS vii
6.5.4. Equality of splicing maps 96
6.6. Construction of the space of global splicing data 98
6.7. Thom-Mather structures on the space of global splicing data 101
6.8. Global splicing map 107
6.9. Projections onto symmetric products 109
Chapter 7. Obstruction bundle 111
7.1. Intro d uc t ion 111
7.2. Infinite-rank obstruction pseudo-bundle 113
7.3. Background obstruction bundle 114
7.4. Equivariant Dirac index bundle 116
7.5. The action of Spinu(4) 117
7.6. Pseudo-bundle over the instanton moduli space with spliced ends 117
7.6.1. Pseudo-bundles and overlap data 119
7.7. Instanton obstruction pseudo-bundle 122
7.7.1. The frame bundles 122
7.7.2. Splicing map 123
7.7.3. Overlap space and overlap maps 124
7.8. Local gluing hypothesis for SO(3) monopoles 126
7.9. Notes on the justification of the local gluing hypothesis 128
7.9.1. Construction of a virtual neighborhood for the moduli space of
SO(3) inonopoles near a top-level singular stratum of SO(3)
monopoles 129
7.9.2. Virtual neighborhoods for the moduli space of anti-self-dual
connections 130
7.9.3. Extrinsic virtual neighborhoods for the moduli space of anti-self-dual
connections and gluing 133
7.9.4. Construction of a virtual neighborhood for the moduli space of
SO(3) monopoles near a lower-level singular stratum of SO(3)
monopoles 135
Chapter 8. Link of an ideal Seiberg-Witten moduli space 137
8.1. Definition of the link of an ideal Seiberg-Witten moduli space 137
8.1.1. The virtual link of an ideal Seiberg-Witten moduli space 137
8.1.2. The link of an ideal Seiberg-Witten moduli space 141
8.1.3. A subspace of the virtual link of an ideal Seiberg-Witten moduli
space 142
8.1.4. Orientations of the link of an ideal Seiberg-Witten moduli space 142
8.1.5. An equality of intersection numbers provided by the SO(3)-monopole
cobordism 143
8.2. Fiber bundle structure of the instanton component of the link of an
ideal Seiberg-Witten moduli space 144
8.3. Boundaries of components of links of ideal Seiberg-Witten moduli
spaces 146
Chapter 9. Cohomology and duality 151
9.1. Introduction 151
9.2. Definitions 152
9.2.1. Subspaces and maps 152
CONTENTS
viii
9.2.2. The incidence locus
9.2.3. Cohomology classes
9.3. Fundamental class of the virtual link of the ideal moduli space of
Seiberg-Witten monopoles
9.4. Computation of the /¿-classes
9.4.1. Geometric representatives and cocycles
9.4.2. Cocycles as pullbacks
9.4.3. Computations of cocycles
9.5. Relative Euler class of the obstruction pseudo-bundle
9.5.1. Euler class of the Seiberg-Witten component of the obstruction
pseudo-bundle
9.5.2. Local Euler class of the instanton component of the obstruction
bundle
9.5.3. Global Euler class of the instanton component of the obstruction
bundle
9.5.4. Relative Euler classes
9.6. Duality and the link of an ideal Seiberg-Witten moduli space
9.6.1. The initial duality
9.6.2. Extension of the cocycles
9.7. Reduction to the subspace BL^
Chapter 10. Computation of the intersection numbers
10.1. Introduction
10.2. Quotient space of BL*
10.2.1. Quotient maps
10.2.2. Construction of the local quotient
10.2.3. Global quotient of BL^
10.3. Homology and cohomology classes of the quotient
10.4. Fiber bundles and pushforwards
10.5. Computations of intersection numbers on Lt s
10.6. Proofs of the main theorems
Chapter 11. Kotschick-Morgan Conjecture
11.1. Cobordisms and reducible connections
11.2. Cohomology classes on the cobordism
11.3. Neighborhoods of gauge-equivalence classes of ideal reducible
connections
11.3.1. Kuranishi model for a neighborhood of a reducible connection
11.3.2. Crude splicing maps
11.3.3. Overlap spaces and maps
11.3.4. Definition of the neighborhood of a gauge-equivalence class of an
ideal reducible connection. ^
11.3.5. Thom-Mather structures on
11.3.6. Global projection map for ™(L)/Sl
11.3.7. Global splicing map on L)/Sl
11.3.8. Obstruction bundle on
11.3.9. Gluing hypothesis
11.4. Cohomology classes on the space of global splicing data
153
155
157
159
159
160
161
166
166
167
172
173
176
176
177
181
187
187
189
190
193
197
200
201
204
206
211
211
212
213
213
214
215
216
217
217
217
218
218
219
CONTKNTS
IX
11.5. Doiiiiition of the link of a gauge-equivalence class of ail ideal reducible?
connection 219
11.G. Computations of the difference term 220
Glossary of Notation 223
Bibliography 227
Index 233
|
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| id | DE-604.BV045468147 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:18:59Z |
| institution | BVB |
| isbn | 9781470414214 |
| language | English |
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| publisher | American Mathematical Society |
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| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Feehan, Paul M. N. 1961- (DE-588)1074475313 aut An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants A. Agrachev, D. Barilari, L. Rizzi Providence, Rhode Island American Mathematical Society [2018] © 2018 xiv, 234, 4 ungezählte Seiten txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society volume 256, number 1226 (second of 6 numbers) Schwefeltrioxid (DE-588)4180436-3 gnd rswk-swf Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd rswk-swf Kobordismus (DE-588)4148171-9 gnd rswk-swf Kobordismus (DE-588)4148171-9 s DE-604 Schwefeltrioxid (DE-588)4180436-3 s Lorentz-Mannigfaltigkeit (DE-588)4299989-3 s Leness, Thomas G. 1967- (DE-588)1176681559 aut Memoirs of the American Mathematical Society volume 256, number 1226 (second of 6 numbers) (DE-604)BV008000141 1226 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030853307&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Feehan, Paul M. N. 1961- Leness, Thomas G. 1967- An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants Memoirs of the American Mathematical Society Schwefeltrioxid (DE-588)4180436-3 gnd Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd Kobordismus (DE-588)4148171-9 gnd |
| subject_GND | (DE-588)4180436-3 (DE-588)4299989-3 (DE-588)4148171-9 |
| title | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants |
| title_auth | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants |
| title_exact_search | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants |
| title_full | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants A. Agrachev, D. Barilari, L. Rizzi |
| title_fullStr | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants A. Agrachev, D. Barilari, L. Rizzi |
| title_full_unstemmed | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants A. Agrachev, D. Barilari, L. Rizzi |
| title_short | An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants |
| title_sort | an so 3 monopole cobordism formula relating donaldson and seiberg witten invariants |
| topic | Schwefeltrioxid (DE-588)4180436-3 gnd Lorentz-Mannigfaltigkeit (DE-588)4299989-3 gnd Kobordismus (DE-588)4148171-9 gnd |
| topic_facet | Schwefeltrioxid Lorentz-Mannigfaltigkeit Kobordismus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030853307&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
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