Volume 15 (2008)

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Compactness and gluing theory for monopoles

Kim A Frøyshov

This book is devoted to the study of moduli spaces of Seiberg-Witten monopoles over spinc Riemannian 4–manifolds with long necks and/or tubular ends. The original purpose of this work was to provide analytical foundations for a certain construction of Floer homology of rational homology 3–spheres; this is carried out in [Monopole Floer homology for rational homology 3–spheres arXiv:08094842]. However, along the way the project grew, and, except for some of the transversality results, most of the theory is developed more generally than is needed for that construction. Floer homology itself is hardly touched upon in this book, and, to compensate for that, I have included another application of the analytical machinery, namely a proof of a "generalized blow-up formula" which is an important tool for computing Seiberg–Witten invariants.

The book is divided into three parts. Part 1 is almost identical to my paper [Monopoles over 4–manifolds containing long necks I, Geom. Topol. 9 (2005) 1–93]. The other two parts consist of previously unpublished material. Part 2 is an expository account of gluing theory including orientations. The main novelties here may be the formulation of the gluing theorem, and the approach to orientations. In Part 3 the analytical results are brought together to prove the generalized blow-up formula.


4–manifolds, rational homology 3–spheres, Seiberg–Witten invariants, monopoles, Floer homology

Mathematical Subject Classification

Primary: 57R58

Secondary: 57R57


Received: 2 November 2006
Revised: 12 August 2008
Published: 30 October 2008

Preface vii
Part I: Compactness
1 Compactness theorems 3
2 Configuration spaces 11
3 Moduli spaces 25
4 Local compactness I 35
5 Local compactness II 43
6 Exponential decay 57
7 Global compactness 75
8 Transversality 83
9 Proofs of Theorems 1.1.1 and 1.1.2 93
Part II: Gluing theory
10 The gluing theorem 99
11 Applications 129
12 Orientations 135
13 Parametrized moduli spaces 159
Part III: An application
14 A generalized blow-up formula 169
A Patching together gauge transformations 181
B A quantitative inverse function theorem 185
C Splicing left or right inverses 189
References 193
Index 197
Kim A Frøyshov
Fakultät für Mathematik
Universität Bielefield
Postfach 10 01 31
D-33501 Bielefeld