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Monopoles and Three-Manifolds
Preface 1. Outlines 2. The Seiberg-Witten equations and compactness 3. Hilbert manifolds and perturbations 4. Moduli spaces and transversality 5. Compactness and gluing 6. Floer homology 7.
The construction of ALE spaces as hyper-Kähler quotients
On decrit la construction d'une famille particuliere de 4-varietes hyper-Kahler: les espaces asymptotiquement localement euclidiens ALE. On decrit une 4-variete de Riemann avec juste une extremite
The Genus of Embedded Surfaces in the Projective Plane
1. Statement of the result The genus of a smooth algebraic curve of degree d in CP is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the
Monopoles and lens space surgeries
Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long
Khovanov homology is an unknot-detector
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
Knots, sutures, and excision
We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots.