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An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg–Witten Invariants
About this Title
Paul M. N. Feehan, Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854-8019 and Thomas G. Leness, Department of Mathematics, Florida International University, Miami, Florida 33199
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 256, Number 1226
ISBNs: 978-1-4704-1421-4 (print); 978-1-4704-4915-5 (online)
DOI: https://doi.org/10.1090/memo/1226
Published electronically: December 27, 2018
Keywords: Anti-self-dual connections,
classification of smooth four-manifolds,
cobordisms,
Donaldson invariants,
gauge theory,
gluing theory,
intersection theory,
Kotschick–Morgan Conjecture,
moduli spaces,
monopoles,
Seiberg–Witten invariants,
stratified spaces,
Uhlenbeck compactification,
Witten’s Conjecture
MSC: Primary 57N13, 57R57, 58D27, 58D29; Secondary 53C07, 53C27, 58J05, 58J20
Table of Contents
Chapters
- Preface
- 1. Introduction
- 2. Preliminaries
- 3. Diagonals of symmetric products of manifolds
- 4. A partial Thom–Mather structure on symmetric products
- 5. The instanton moduli space with spliced ends
- 6. The space of global splicing data
- 7. Obstruction bundle
- 8. Link of an ideal Seiberg–Witten moduli space
- 9. Cohomology and duality
- 10. Computation of the intersection numbers
- 11. Kotschick–Morgan Conjecture
- Glossary of Notation
Abstract
We 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 moduli space of $\mathrm {SO}(3)$ monopoles (Feehan and Leness, $\rm PU(2)$ monopoles. I. Regularity, Uhlenbeck compactness, and transversality, 1998). In this monograph, we prove — modulo a gluing theorem which is an extension of our earlier work in $\rm PU(2)$ monopoles. III: Existence of gluing and obstruction maps (arXiv:math/9907107)— that these intersection pairings can be expressed in terms of topological data and Seiberg–Witten invariants of the four-manifold. Our proofs that the $\mathrm {SO}(3)$-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze (Superconformal invariance and the geography of four-manifolds, 1999; Four-manifold geography and superconformal symmetry, 1999) and Witten’s Conjecture (Monopoles and four-manifolds, 1994) in full generality for all closed, oriented, smooth four-manifolds with $b_1=0$ and odd $b^+\ge 3$ appear in Feehan and Leness, Superconformal simple type and Witten’s conjecture (arXiv:1408.5085) and $\mathrm {SO}(3)$ monopole cobordism and superconformal simple type (arXiv:1408.5307).- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
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