An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg-Witten Invariants.
- 1st ed.
- 1 online resource (254 pages)
- Memoirs of the American Mathematical Society Series ; v.256 .
- Memoirs of the American Mathematical Society Series .
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 -- Chapter 2. Preliminaries -- 2.1. 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 -- Chapter 3. Diagonals of symmetric products of manifolds -- 3.1. 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 -- 4.4.4. The projection maps. 4.5. Construction of the families of locally flattened metrics -- 4.6. Normal bundles of strata of \Sym^( ) -- 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 -- 5.5.3. Symmetric group actions and quotients -- 5.6. 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 -- 6.4.1. The standard splicing map -- 6.4.2. Construction of the crude splicing map -- 6.4.3. Properties of the crude splicing map -- 6.5. Overlap spaces and maps -- 6.5.1. The overlap space -- 6.5.2. The upwards overlap map -- 6.5.3. Downwards overlap map -- 6.5.4. Equality of splicing maps -- 6.6. Construction of the space of global splicing data -- 6.7. Thom-Mather structures on the space of global splicing data -- 6.8. Global splicing map -- 6.9. Projections onto symmetric products -- Chapter 7. Obstruction bundle -- 7.1. Introduction -- 7.2. Infinite-rank obstruction pseudo-bundle -- 7.3. Background obstruction bundle -- 7.4. Equivariant Dirac index bundle. 7.5. The action of \Spinu(4) -- 7.6. Pseudo-bundle over the instanton moduli space with spliced ends -- 7.6.1. Pseudo-bundles and overlap data -- 7.7. Instanton obstruction pseudo-bundle -- 7.7.1. The frame bundles -- 7.7.2. Splicing map -- 7.7.3. Overlap space and overlap maps -- 7.8. Local gluing hypothesis for \SO(3) monopoles -- 7.9. Notes on the justification of the local gluing hypothesis -- 7.9.1. Construction of a virtual neighborhood for the moduli space of \SO(3) monopoles near a top-level singular stratum of \SO(3) monopoles -- 7.9.2. Virtual neighborhoods for the moduli space of anti-self-dual connections -- 7.9.3. Extrinsic virtual neighborhoods for the moduli space of anti-self-dual connections and gluing -- 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 -- Chapter 8. Link of an ideal Seiberg-Witten moduli space -- 8.1. Definition of the link of an ideal Seiberg-Witten moduli space -- 8.1.1. The virtual link of an ideal Seiberg-Witten moduli space -- 8.1.2. The link of an ideal Seiberg-Witten moduli space -- 8.1.3. A subspace of the virtual link of an ideal Seiberg-Witten moduli space -- 8.1.4. Orientations of the link of an ideal Seiberg-Witten moduli space -- 8.1.5. An equality of intersection numbers provided by the \SO(3)-monopole cobordism -- 8.2. Fiber bundle structure of the instanton component of the link of an ideal Seiberg-Witten moduli space -- 8.3. Boundaries of components of links of ideal Seiberg-Witten moduli spaces -- Chapter 9. Cohomology and duality -- 9.1. Introduction -- 9.2. Definitions -- 9.2.1. Subspaces and maps -- 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 \bB\bL^_ -- Chapter 10. Computation of the intersection numbers -- 10.1. Introduction -- 10.2. Quotient space of \bB\bL^_ -- 10.2.1. Quotient maps -- 10.2.2. Construction of the local quotient -- 10.2.3. Global quotient of \bB\bL^_ -- 10.3. Homology and cohomology classes of the quotient -- 10.4. Fiber bundles and pushforwards -- 10.5. Computations of intersection numbers on \bL_ -- 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 ̃\sU^_( )/ ¹ -- 11.3.6. Global projection map for ̃\sU^_( )/ ¹ -- 11.3.7. Global splicing map on ̃\sU^_( )/ ¹ -- 11.3.8. Obstruction bundle on ̃\sU^_( )/ ¹ -- 11.3.9. Gluing hypothesis -- 11.4. Cohomology classes on the space of global splicing data. 11.5. Definition of the link of a gauge-equivalence class of an ideal reducible connection -- 11.6. Computations of the difference term -- Glossary of Notation -- Bibliography -- Index -- Back Cover.
The authors prove an analogue of the Kotschick-Morgan Conjecture in the context of \mathrm monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the \mathrm-monopole cobordism. The main technical difficulty in the \mathrm-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible \mathrm monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of \mathrm monopoles. In this monograph, the authors prove--modulo a gluing theorem which is an extension of their earlier work--that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. Their proofs that the \mathrm-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze and Witten's Conjecture in full generality for all closed, oriented, smooth four-manifolds with b_1=0 and odd b^+\ge 3 appear in earlier works.