We investigate Seiberg-Witten theory in the presence of real structures. Certain conditions are obtained so that integer valued real Seiberg-Witten invariants can be defined. In general we study properties of the real Seiberg-Witten projection map from the point of view of Fredholm map degrees.
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48 results for “Seiberg-Witten map”
New framework for Seiberg-Witten map on non-compact 4-manifolds.
problem L^2 self-dual forms on non-compact 4-manifolds cannot exhaust d^+ images.
method Functional analytic framework for Seiberg-Witten map.
result Construction of new functional analytic framework.
Study shows symplectic mapping groups of K3 surfaces are infinitely generated.
problem Understanding symplectic mapping class groups of K3 surfaces.
method Uses Kronheimer's approach and Seiberg-Witten invariants.
result Symplectic mapping class groups of many K3 surfaces are infinitely generated.
A surgery formula for a specific 4-manifold invariant.
problem Computing the Casson-Seiberg-Witten invariant of a specific 4-manifold.
method Surgery formula and application to mapping tori.
result Computed the invariant for mapping tori of 3-manifolds.
The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.
This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…
Compact moduli space shown for Seiberg-Witten on flat scalar curvature manifold.
problem Compactification of Seiberg-Witten moduli space on manifolds with flat scalar curvature.
method Analysis of Seiberg-Witten equations on non-compact manifolds with periodic ends, scalar curvature zero, and vanishing cohomologies.
result Moduli space is compact under given conditions.
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
We prove integral rigidity for Seiberg-Witten invariants of 4-manifolds with specific hypersurfaces.
problem Integral rigidity of Seiberg-Witten invariants in 4-manifolds with non-separating hypersurfaces.
method Floer theoretic conditions and interplay between irreducible and reducible solutions to Seiberg-Witten equations.
result Sum of Seiberg-Witten invariants is determined cohomologically for specific 4-manifolds.
Seiberg-Witten Floer spectra for math.GT
New spectral construction for 3-manifolds with non-zero first Betti number.
problem Constructing a new spectral sequence for 3-manifolds with non-zero first Betti number.
method Defining Bauer-Furuta maps on generalized Seiberg-Witten Floer spectra for suitable 3-manifolds.
result Computing examples of these spectra and maps.
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom.…
The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.
problem Examining exotic diffeomorphisms in 4-manifolds with a specific topological invariant.
method Utilizes Seiberg-Witten invariants for 1-parameter families of 4-manifolds and a gluing formula for connected sums.
result Proves that the mapping class group of certain 4-manifolds is not finitely generated and surjects to Z^∞.
We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the S…
We establish a canonical gluing procedure for Seiberg-Witten monopoles on the two pieces of a closed, oriented 4-manifold X which is split along a 3-dimensional closed, oriented submanifold. We only assume that the (unperturbed) character variety is Kuranishi-smooth and the limiting maps are transversal -- then we will…
Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.
problem Analyzing moduli spaces of Seiberg-Witten equations on manifolds with boundary.
method General regularity theorem, strong unique continuation principle, and gluing theorem for Dirac operators; smoothness of restriction map.
result Proves moduli spaces are Hilbert manifolds and have semi-infinite-dimensionality properties.
The paper proves rigidity of Seiberg-Witten invariants for spin 4-manifold families.
problem Rigidity of Seiberg-Witten invariants for spin 4-manifold families.
method Mechanism of rigidity theorem and 10/8-type inequality.
result Existence of non-smoothable topological families of 4-manifolds.
This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gau…
We give an alternative proof of the mod vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when . Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric. Furt…
We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.
Study defines a new invariant for foliated manifolds, linking index of Dirac operator to cohomology.
problem Defining a new invariant for foliated manifolds.
method Using transversal Seiberg-Witten map and finite dimensional approximation.
result Estimates and vanishing conditions for the index of transverse Dirac operator.
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
The paper constrains families of smooth 4-manifolds using Seiberg-Witten invariants.
problem Understanding the topology of families of smooth 4-manifolds.
method Finite dimensional approximation of the Seiberg-Witten monopole map.
result Constructs examples of continuous -actions and shows non-smoothability.
The paper calculates Seiberg-Witten invariants and finds non-symplectic diffeomorphisms.
problem Calculating Seiberg-Witten invariants for families of 4-manifolds.
method Extending Kronheimer-Mrowka's result to family setting, using gluing formula and monopole Floer (co)homology.
result Establishes a large family of simply-connected 4-manifolds with nontrivial fundamental group of diffeomorphisms.
New equations use Pin(2) symmetry to study spinor and connection solutions.
problem Study of spinor and connection solutions on 4-manifolds.
method Define and analyze Seiberg-Witten-like equations with Rarita-Schwinger operators.
result Moduli space of solutions is non-compact and non-empty under certain conditions.
Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.
problem Constraints on the topology of Lefschetz fibrations with 4D fibers.
method Using the family Bauer-Furuta invariant and framed bordism class of 1D Seiberg-Witten moduli spaces.
result New obstructions to the smooth isotopy of compositions of Dehn twists.
Analyzes Seiberg-Witten theory on 4-manifolds with periodic ends, proving invariants and compactness.
problem Analyzing Seiberg-Witten theory on 4-manifolds with periodic ends.
method Using Taubes criteria and Fredholm operators, proving Hodge type decompositions and isomorphisms.
result Defines a Seiberg-Witten type cohomotopy invariant for 4-manifolds with periodic ends.
Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.
problem Homotopy coherent Nielsen realization problem for Dehn twists on 4-manifolds
method Using family Seiberg-Witten theory
result Failure of the classical Nielsen realization problem in K3-type 4-manifolds
In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prov…
New method constructs multi-monopoles on mapping tori.
problem Understanding wall-crossing in multi-monopole counts.
method Adiabatic limit theorem to construct multi-monopoles.
result First explicit constructions of multi-monopoles in various chambers.
The homology of 4-manifold moduli spaces can be infinitely generated.
problem The homology of moduli spaces of 4-manifolds is not finitely generated.
method The proof uses characteristic classes from Seiberg-Witten theory.
result There exist simply-connected closed smooth 4-manifolds whose homology groups are not finitely generated.
Three theorems on K3 surfaces' diffeomorphism and homeomorphism groups.
problem Properties of diffeomorphism and homeomorphism groups of K3 surfaces.
method Seiberg-Witten theory and global Torelli theorem.
result The natural map π₀(Diff(K3)) → Aut(H²(K3; ℤ)) has a section over its image.
Formulae prove equivalence of two invariants on specific 4-manifolds.
problem Equivalence of two invariants on homology .
method Surgery and excision formulas for Furuta-Ohta invariant .
result Computes of certain families of manifolds.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
Formula for combining Seiberg-Witten invariants of 4-manifolds.
problem Combining invariants of 4-manifolds obtained by connected sum.
method Proved a gluing formula for families Seiberg-Witten invariants.
result Formula expresses invariants of connected sum in terms of one summand.
Positive scalar curvature metrics on cobordisms yield only reducible Seiberg-Witten solutions.
problem Counting solutions to Seiberg-Witten equations on cobordisms.
method Constructing families of metrics with positive scalar curvature.
result Irreducible solutions are absent when positive scalar curvature metrics are used.
Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of -equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…
In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on…
We show a non-existence result for some class of equivariant maps between sphere bundles over tori. The notion of equivariant KO-degree is used in the proof. As an application to Seiberg-Witten theory, for a connected closed oriented spin 4-manifold with indefinite intersection form, we have a new bound of the second B…
Study defines PFH cobordism maps from Lefschetz fibrations.
problem Indirectly defined PFH cobordism maps using Seiberg-Witten theory.
method Defined PFH cobordism maps using holomorphic curves.
result Computed PFH cobordism maps for specific cases.
We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relat…
Formula calculates Seiberg-Witten invariants after surgery.
problem Calculating Seiberg-Witten invariants for surgeries on 4-manifolds.
method Surgery formula for Seiberg-Witten invariants of 4-manifolds.
result Formula expresses invariants of the manifold after surgery.
On finite energy monopoles on math.DG
Finite energy solutions classified for Seiberg-Witten equations on complex plane and Riemann surface.
problem Classifying solutions to Seiberg-Witten equations with finite energy.
method Established a classification theorem for solutions on with finite analytic energy.
result Finite energy solutions correspond to polynomial maps from to .
On the compactness problem for a family of generalized Seiberg-Witten equations in dimension threemath.DG
Abstract: Proves compactness theorem for generalized Seiberg-Witten equations in 3D.
problem Compactness of solutions to generalized Seiberg-Witten equations in 3D.
method Abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three.
result Recovers and extends known compactness theorems for stable flat connections and Seiberg-Witten equations.
New formula connects Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds.
problem Understanding invariants of families of 4-manifolds.
method Cohomological formula linking Bauer-Furuta and Seiberg-Witten invariants.
result New divisibility properties of families Seiberg-Witten invariants.
This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli space of solutions to t…
The Seiberg-Witten equations for multiple-spinors on manifolds with definite intersection formsmath.GT
Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.
problem Donaldson's Diagonalization Theorem
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple spinors
result Proof of Donaldson's Diagonalization Theorem