Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.
problem Investigating invariants of 4-manifolds under group actions and Galois coverings.
method Functorial approach to equivariant invariants and study in Galois covering situations.
result Ordinary invariants of quotients are determined by equivariant invariants of the covering manifold.
Study BF invariants using simple type concepts.
problem Understanding Bauer--Furuta invariants for 4-manifolds.
method Extend simple type concept to BF blowup and BF homogeneous types, prove gluing formulae and adjunction inequality.
result Determine Bauer--Furuta invariant of a specific 4-manifold and give constraints on gluing decompositions.
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.
New inequality for links in 3D using 4D invariants.
problem Understanding links in 3D using 4D invariants.
method Using a Bauer--Furuta-type invariant for 4-manifolds with contact boundary.
result Generalized Thurston--Bennequin inequality for links in S3. Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. Novel construction of Bauer--Furuta invariant using sheaves of spectra.
problem Constructing the Bauer--Furuta invariant without finite-dimensional approximations.
method Using sheaves of spectra and Borel--Moore homology, avoiding approximations.
result Defines the shriek functors and Thom spectra for index calculations.
Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.
problem Characterizing exotic diffeomorphisms on Milnor fibers.
method Compared family relative Bauer--Furuta invariants of mapping tori.
result Boundary Dehn twist is an exotic diffeomorphism under given conditions.
Established a stable cohomotopy refinement for Pin(2) monopole invariants.
problem Refining monopole invariants for Pin(2) structures.
method Adapted Bauer-Furuta's Seiberg-Witten refinement method to Pin(2) monopole invariants.
result Connected sum formula for the refined invariants.
In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framewor…
S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable loc…
The study explores stable diffeomorphism groups in 4-manifolds using localisation and invariants.
problem Understanding stable diffeomorphism groups in 4-manifolds.
method Localisation of n-manifolds, inverting connected sum construction, using Bauer--Furuta invariants.
result K3-stable Bauer--Furuta invariants determine S^2xS^2-stable invariants.
Study cohomotopy classes for 4-manifolds using complex spin structures.
problem Understanding cohomotopy classes for families of 4-manifolds with complex spin structures.
method Using Bauer--Furuta invariants in parametrised stable homotopy theory.
result Definition of characteristic cohomotopy classes on Thom spectra.
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the res…
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.
problem Understanding sliceness of knots and symplectic embeddings in 4-manifolds.
method An adjunction inequality for embedded surfaces in 4-manifolds with contact boundaries.
result Infinitely many knots are topologically H-slice but not smoothly H-slice in certain 4-manifolds.
Kronheimer and Mrowka constructed a variant of Seiberg-Witten invariants for a 4-manifold X with contact boundary in 1997. Using Furuta's finite dimensional approximation, we refine this invariant in the case H1(X,∂X;R)=0.
New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on X1#X2 is not isotopic to identity if determinants r1,r2 are odd. We show how the families Seiberg-Witten invariants of a family of smooth 4-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the fami…
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-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative i…
New invariant connects symplectic fillings and contact structures.
problem Understanding symplectic fillings and contact structures.
method Floer homotopy theory and KO-cohomology.
result Constraint on symplectic fillings using KO-cohomology.
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.
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
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.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S1-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…
We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estima…
Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.
problem Positive scalar curvature metrics on specific 4-manifolds.
method Relative Bauer-Furuta-type invariant on periodic-end 4-manifolds.
result Obstructions to positive scalar curvature metrics on rational homology S1imesS3. We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…
We obtain constraints on the topology of families of smooth 4-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's 10/8 theorem. As an appli…
For closed manifolds endowed with a Riemannian foliation of codimension 4, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that Hb1(M)∩H1(M,Z) is a lattice of Hb1(M), we can define a…
New example of non-smooth isotopy after stabilization in 4-manifolds.
problem Non-smooth isotopy of Dehn twist after stabilization.
method Pin(2)-equivariant Bauer-Furuta invariant.
result Dehn twist is not smoothly isotopic even after stabilization.
On a smooth closed oriented 4-manifold M with a smooth action by a compact Lie group G, we define a G-monopole class as an element of H2(M;Z) which is the first Chern class of a G-equivariant Spinc structure which has a solution of the Seiberg-Witten equations for any G-invariant Riemannian metri…
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
Diagonalizes tautological classes of definite 4-manifolds.
problem Diagonalizing tautological classes of definite 4-manifolds.
method Families version of Bauer-Furuta cohomotopy refinement of Seiberg-Witten theory.
result Completely determined tautological rings of CP2 and CP2#CP2. Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
The θ invariant encompasses the Rozansky-Overbay invariant.
problem None explicitly stated in the abstract.
method Generalization of the Rozansky-Overbay invariant using the θ invariant. result The θ invariant recovers the Rozansky-Overbay invariant. Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces J2+-invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points. result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
New invariant CWR for alternating links is stronger than existing invariants.
problem Developing a stronger invariant for alternating links.
method Introducing CWR invariant as an array of two-variable polynomials. result The CWR invariant is stronger than classical invariants like HOMFLYPT and Kauffman polynomials. Defines knot concordance invariant using instanton homology and Donaldson invariants.
problem Knot concordance and its classification.
method Defines an invariant φ for knots in the 3-sphere using Donaldson invariants and Floer's instanton homology. result The invariant φ coincides with a special case of an invariant defined by Froyshov. Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
New family of knots with epsilon invariant nonzero despite Upsilon and phi being zero.
problem Comparing smooth concordance invariants.
method Building an infinite family of knots.
result Found knots with epsilon invariant nonzero but Upsilon and phi zero.