Originating 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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Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

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A b s t r a c t . We show how the new invariants of 4-manifolds resulting from the Seiberg-Witten monopole equation lead quickly to a proof of the 'Thom conjecture'. Statement of the resultThe genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,8,10]. In this note we confirm the conjecture. Very recently, Seiberg and Witten [11,12,13] introduced new invariants of 4-manifolds, closely related to Donaldson's polynomial invariants [2], but in many respects much simpler to work with. The new techniques have led to more elementary proofs of many theorems in the area. Given the monopole equation and the vanishing theorem which holds when the scalar curvature is positive (something which was pointed out by Witten), the rest of the argument presented here is not hard to come by. A slightly different proof of the Theorem, based on the same techniques, has been found by Morgan, Szabo and Taubes.It is also possible to prove a version of Theorem 1 for other complex surfaces, without much additional work. This and various other applications will be treated in a later paper, with joint authors. The monopole equation and the Seiberg-Witten invariantsLet X be an oriented, closed Riemannian 4-manifold. Let a spin c structure on X be given. We write c for the spin c structure and write W + = W