| Description |
1 online resource (254 pages) |
| Series |
Memoirs of the American Mathematical Society Ser. ; v. 256 |
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Memoirs of the American Mathematical Society Ser
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| Contents |
Cover; Title page; Preface; Acknowledgments; Chapter 1. Introduction; 1.1. Summary of main results; 1.2. Outline of the argument; 1.2.1. Problem of overlaps; 1.2.2. Overlap space and overlap maps; 1.2.3. Associativity of splicing maps; 1.2.4. Instanton moduli space with spliced ends; 1.2.5. Space of global splicing data; 1.2.6. Definition of link of a subspace of a moduli space of ideal Seiberg-Witten monopoles; 1.2.7. Computation of intersection numbers with the link of the moduli space of ideal Seiberg-Witten monopoles; 1.3. Kotschick-Morgan Conjecture; 1.4. Outline of the monograph |
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Chapter 2. Preliminaries2.1. The moduli space of \SO(3) monopoles; 2.1.1. Clifford modules; 2.1.2. \SO(3) monopoles; 2.2. Stratum of anti-self-dual or zero-section solutions; 2.3. Strata of Seiberg-Witten or reducible solutions; 2.3.1. Seiberg-Witten monopoles; 2.3.2. Seiberg-Witten invariants; 2.3.3. Reducible \SO(3) monopoles; 2.3.4. Circle actions; 2.3.5. The virtual normal bundle of the Seiberg-Witten moduli space; 2.4. Cohomology classes on the moduli space of \SO(3) monopoles; 2.5. Donaldson invariants; 2.6. Links and the cobordism |
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Chapter 3. Diagonals of symmetric products of manifolds3.1. Definitions; 3.1.1. Subgroups of the symmetric group; 3.1.2. Definition of the diagonals; 3.1.3. Strata of the symmetric product; 3.2. Incidence relations among diagonals and strata; 3.3. Normal bundles of diagonals and strata; 3.4. Enumeration of the strata; Chapter 4. A partial Thom-Mather structure on symmetric products; 4.1. Introduction; 4.2. Diagonals in products of \RR⁴; 4.3. Families of metrics; 4.4. Overlap maps; 4.4.1. The downwards overlap map; 4.4.2. The upwards overlap map; 4.4.3. Commuting overlap maps |
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4.4.4. The projection maps4.5. Construction of the families of locally flattened metrics; 4.6. Normal bundles of strata of \Sym̂{ℓ}(); 4.7. The tubular distance function; 4.8. Decomposition of the strata; Chapter 5. The instanton moduli space with spliced ends; 5.1. Introduction; 5.2. Connections over the four-dimensional sphere; 5.3. Strata containing the product connection; 5.3.1. Tubular neighborhoods; 5.4. The splicing map with the product connection over \RR⁴; 5.5. Composition of splicing maps; 5.5.1. Definition of the overlap data; 5.5.2. Equality of splicing maps |
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5.5.3. Symmetric group actions and quotients5.6. The spliced end of the instanton moduli space; 5.7. Tubular neighborhoods of the instanton moduli space with spliced ends; 5.8. Isotopy of the spliced end of the instanton moduli space; 5.9. Properties of the instanton moduli space with spliced ends; Chapter 6. The space of global splicing data; 6.1. Introduction; 6.2. Splicing data; 6.2.1. Background pairs; 6.2.2. Riemannian metrics; 6.2.3. Frame bundles; 6.2.4. Group actions on the frame bundles; 6.2.5. Space of splicing data; 6.3. The flattening map on pairs; 6.4. The crude splicing map |
| Summary |
The authors prove an analogue of the Kotschick-Morgan Conjecture in the context of \mathrm{SO(3)} monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the \mathrm{SO(3)}-monopole cobordism. The main technical difficulty in the \mathrm{SO(3)}-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible \mathrm{SO(3)} monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the modu |
| Notes |
6.4.1. The standard splicing map |
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Print version record |
| Subject |
Cobordism theory.
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Four-manifolds (Topology)
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Seiberg-Witten invariants.
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Cobordism theory
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Four-manifolds (Topology)
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Seiberg-Witten invariants
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| Form |
Electronic book
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| Author |
Leness, Thomas G
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| ISBN |
9781470449155 |
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1470449153 |
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