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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 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.
Gauge theory for embedded surfaces, II
This paper is the second in a series of two, aimed at developing results about the topology of embedded surfaces Σ in a 4-manifold X using some new YangMills moduli spaces associated to such pairs
Knot homology groups from instantons
For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.