Monopoles and Three-Manifolds

  title={Monopoles and Three-Manifolds},
  author={Peter B. Kronheimer and Tomasz S. Mrowka},
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. Cobordisms and invariance 8. Non-exact perturbations 9. Calculations 10. Further developments References Glossary of notation Index. 
Given a three-manifold with b1 = 1 and a nontorsion spin c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodicExpand
The Seiberg–Witten equations and the Weinstein conjecture
Let M denote a compact, oriented 3–dimensional manifold and let a denote a contact 1–form on M; thus a∧da is nowhere zero. This article proves that the vector field that generates the kernel of daExpand
Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric andExpand
Equivariant Floer theory and double covers of three-manifolds.
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, whichExpand
Lectures on monopole Floer homology
These lecture notes are a friendly introduction to monopole Floer homology. We discuss the relevant differential geometry and Morse theory involved in the definition. After developing the relationExpand
Superconformal simple type and Witten's conjecture
Abstract Let X be a smooth, closed, connected, orientable four-manifold with b 1 ( X ) = 0 and b + ( X ) ≥ 3 and odd. We show that if X has Seiberg–Witten simple type, then the SO ( 3 ) -monopoleExpand
Unfolded Seiberg-Witten Floer spectra, II: Relative invariants and the gluing theorem
We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifoldsExpand
Embedded contact homology and Seiberg-Witten Floer cohomology I
This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. ThisExpand
Seiberg-Witten equation on a manifold with rank-2 foliation
Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariantExpand
Floer theory and its topological applications
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, andExpand


Review : Monopoles and three - manifolds by Peter Kronheimer and Tomasz Mrowka ( PDF )
  • 2009