eBook ISBN:  9781470449155 
Product Code:  MEMO/256/1226.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470449155 
Product Code:  MEMO/256/1226.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 256; 2018; 228 ppMSC: Primary 57; 58; Secondary 53
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 fourmanifolds 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 lowerlevel strata of the Uhlenbeck compactification of the moduli space of \(\mathrm{SO(3)}\) 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 fourmanifold. Their proofs that the \(\mathrm{SO(3)}\)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 fourmanifolds with \(b_1=0\) and odd \(b^+\ge 3\) appear in earlier works.

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


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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 fourmanifolds 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 lowerlevel strata of the Uhlenbeck compactification of the moduli space of \(\mathrm{SO(3)}\) 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 fourmanifold. Their proofs that the \(\mathrm{SO(3)}\)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 fourmanifolds with \(b_1=0\) and odd \(b^+\ge 3\) appear in earlier works.

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