## Monopoles and Three-ManifoldsOriginating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in three- and four-dimensional geometry and topology. This 2007 book provides a comprehensive treatment of Floer homology, based on the Seiberg–Witten monopole equations. After first providing an overview of the results, the authors develop the analytic properties of the Seiberg–Witten equations, assuming only a basic grounding in differential geometry and analysis. The Floer groups of a general three-manifold are then defined and their properties studied in detail. Two final chapters are devoted to the calculation of Floer groups and to applications of the theory in topology. Suitable for beginning graduate students and researchers, this book provides a full discussion of a central part of the study of the topology of manifolds. |

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### Contents

1 | |

II The SeibergWitten equations and compactness | 84 |

III Hilbert manifolds and perturbations | 134 |

IV Moduli spaces and transversality | 195 |

V Compactness and gluing | 274 |

VI Floer homology | 375 |

VII Cobordisms and invariance | 449 |

VIII Nonexact perturbations | 590 |

IX Calculations | 634 |

X Further developments | 721 |

779 | |

Glossary of notation | 785 |

792 | |

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### Common terms and phrases

2-plane ﬁeld 4-dimensional 4-manifold b+(X blow-up boundary-obstructed boundary-stable bounded broken trajectories bundle canonical Chern class cobordism coefﬁcients cohomology cokernel compact components conﬁguration space converges Corollary corresponding critical points cup product cylinder define deﬁned Deﬁnition denote diffeomorphism differential dimension Dirac operator eigenvalues element ﬁber ﬁnite ﬁnite-dimensional ﬁrst Floer groups Floer homology ﬂow follows formula Fredholm Fredholm operator gauge group gauge transformation grad grading Hilbert manifold homology groups homology orientation homotopy class inﬁnite invariant irreducible isomorphism kernel L2 norm Lemma linear manifold with boundary metric moduli space Morse complex Morse function multiplication neighborhood non-zero obtain pair parametrized path perturbation quotient reducible restriction result Riemannian Seiberg–Witten equations smooth Sobolev solutions spectral spinc structure submanifold Subsection subspace Suppose surjective tangent Theorem topological torsion transverse trivial vector ﬁeld write zero