Two Pin(2)-equivariant Floer homologies are shown to be equivalent.
problem Equivalence of Pin(2)-equivariant Floer homologies.
method Comparison of definitions by Lin and Manolescu.
result Pin(2)-equivariant monopole Floer homology is isomorphic to Seiberg-Witten Floer homology.
Remark on Pin(2)-equivariant Floer homology and its connections.
problem Understanding the relationship between Pin(2)-equivariant Floer homology and other invariants.
method Analyzing the monopole Frøyshov invariant and Involutive Heegaard Floer correction terms.
result Identifying the only interesting correction terms as coming from subgroups Z/4, S^1, and Pin(2).
Computes Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds.
problem Computing Pin(2)-equivariant monopole Floer homology for specific 3-manifolds.
method Uses Heegaard Floer/monopole Floer lattice complex and ranks of monopole Floer homology groups.
result Pin(2)-equivariant monopole Floer homology can be determined by ranks of monopole Floer homology groups.
Computes Pin(2)-equivariant Seiberg-Witten Floer homology for Seifert fibrations.
problem Homology cobordism between Seifert fiber spaces and integral homology spheres.
method Uses Heegaard Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Proves Manolescu's conjecture and identifies new homology cobordism obstructions.
Research examines triangulations and homology cobordism groups.
problem Understanding the three-dimensional homology cobordism group.
method Local equivalence methods from Pin(2)-equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
result Review of known results and new insights into the group.
Floer theory applied to manifold surgery and triangulation.
problem Understanding manifold surgeries and triangulations.
method Floer homology and Pin(2)-equivariant Seiberg-Witten Floer homology.
result Floer stable homotopy types and Pin(2)-equivariant Seiberg-Witten Floer homology provide insights into the triangulation conjecture.
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…
Formula connects knot invariant to Lefschetz number, proving special case for Seifert solids.
problem Establishing a formula relating Miyazawa's knot invariant to Lefschetz number.
method Using monopole Floer homology with Pin(2)-equivariant perturbations and integer coefficients.
result Proves ∣deg∣=1 for certain 2-knots in S4 with specific Seifert solid properties. Study non-formality in Pin(2)-monopole Floer homology.
problem Understanding non-formality in Pin(2)-monopole Floer homology.
method Introduced a natural A-infinity structure and computed Massey products.
result Explicit descriptions and computations of Pin(2)-monopole Floer homology.
Proves exact triangle for Pin(2)-monopole Floer homology.
problem Existence of exact triangle in Pin(2)-monopole Floer homology.
method Using specific Dehn surgeries and elementary cobordism.
result Describes invariants of homology spheres.
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer K_G-split homology sphere; thi…
Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres $\pmΣ(2,3,6k\…
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spinc structure isomorphic to its conjugate, we define the counterpart in this…
Study of Pin(2)-monopole Floer homology under connected sums.
problem Understanding Pin(2)-monopole Floer homology behavior under connected sums.
method Constructing a partially defined A-infinity module structure and using Eilenberg-Moore spectral sequences.
result Identify the Floer chain complex of a connected sum via an A-infinity tensor product of modules.
Research provides obstructions to Stein fillings of certain rational homology spheres.
problem Obstructing Stein fillings of specific rational homology spheres.
method Uses Pin(2)-monopole Floer homology.
result Provides obstructions to the intersection forms of Stein fillings.
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.
Proof outlines existence of non-triangulable manifolds in high dimensions.
problem Existence of non-triangulable manifolds in dimensions >4.
method Homology cobordism invariants from Pin(2) symmetry of Seiberg-Witten equations.
result Existence of non-triangulable manifolds proven in dimensions >4.
Monopole Floer homology links Rokhlin invariants in 3-manifolds.
problem Understanding the relationship between Pin(2)-monopole Floer homology and Rokhlin invariants. method Using mod 2 index theory, the bar version of Pin(2)-monopole Floer homology is determined by triple cup products and Rokhlin invariants. result The Pin(2)-monopole Floer homology of a 3-manifold is determined by its Rokhlin invariants. New conditions prevent non-trivial relations in local equivalence group.
problem Understanding non-trivial relations in homology cobordism groups.
method Filtered instanton Floer homology and rs-invariants. result Certain conditions prevent non-trivial relations in the local equivalence group.
Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.
problem Understanding cyclic group actions on spin 4-manifolds with boundary.
method Using Seiberg-Witten equations and lattice constructions, define equivariant refinements of the invariant κ.
result Equivariant relative 10/8-ths type inequalities for spin equivariant cobordisms between rational homology spheres.
New method for Lagrangian Floer homology groups using flow trees.
problem Computing equivariant Lagrangian Floer homology.
method Constructing and exploiting an A-infinity module structure on the Floer complex.
result Established constructions of equivariant Lagrangian Floer homology groups.
Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.
problem Understanding the action of symmetries on knot Floer homology.
method Relating knot Floer homology to Heegaard Floer homology via equivariant surgeries.
result Identify the action of the involution on Heegaard Floer homology with an action on knot Floer homology.
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
Develops Floer theory for 3-manifold covers using equivariant structures.
problem Calculating Steenrod operations and Heegaard Floer homology of 3-manifold covers.
method Equivariant Floer theory with Z/2-actions and localization. result Localization to twisted Floer cohomology for invariant sets.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
Study calculates Floer homology for binary polyhedral spaces.
problem Calculating Floer homology for specific polyhedral spaces.
method Equivariant instanton Floer homology, modified algebraic construction.
result Equivariant instanton Floer homology values for binary polyhedral spaces.
For a rational homology 3-sphere Y with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a dua…
Two Seiberg-Witten homologies are proven equivalent.
problem Equivalence of Seiberg-Witten homologies.
method Isomorphism proof between monopole Floer homology and Seiberg-Witten spectrum.
result Monopole Floer homology is isomorphic to Seiberg-Witten spectrum.
Real Heegaard Floer theory shown to be natural and invariant.
problem Defining and proving naturality of real Heegaard Floer homology.
method Defined and proved naturality of real Heegaard Floer homology and other related theories.
result Real Heegaard Floer homology is shown to be natural and admits an action of the equivariant mapping class group.
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Paper constructs new equivariant Floer cohomology and proves invariance properties.
problem Invariance properties of spectral sequences in symplectic Khovanov homology and Heegaard Floer homology.
method Flexible construction of equivariant Floer cohomology with finite group action.
result Proves invariance properties of spectral sequences and introduces new concordance homomorphism.
We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.
Study of involutive monopole Floer homology and Khovanov homology in characteristic two.
problem Investigate involutive monopole Floer homology and its relation to Khovanov homology.
method Use Seiberg-Witten theory and spectral sequences to relate homology theories.
result Prove the existence of a spectral sequence converging to involutive monopole Floer homology.
Proves an equivariant version of Heegaard Floer link surgery formula.
problem Equivariant knot surgery in S3. method Naturality theorem for bordered modules.
result Kernel of forgetful map contains a Z∞-summand. We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
New instanton homology theories for 3-manifolds and bundles.
problem Defining instanton homology for a class of 3-manifolds and bundles.
method Functorial construction for 4-manifold cobordisms, algebraic construction of homology theories for dg-modules.
result Isomorphic to Floer's instanton homology for admissible bundles, calculates I∞ in all cases it is defined. Contact gluing maps are shown to be equivalent in sutured Floer homology.
problem Equivalence of contact gluing maps in sutured Floer homology.
method Established a conjecture by Zarev, showing the contact gluing map is equivalent to the sutured Floer homology gluing map.
result Contact gluing maps are equivalent in sutured Floer homology.
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 construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first…
New invariant recovers known contact element and considers finite coverings.
problem Defining and studying new contact invariants in Seiberg-Witten Floer spectra.
method Cohomotopy set of Seiberg-Witten Floer spectrum, equivariant Borel cohomology.
result New invariant recovers known contact element and considers finite coverings.
New homological results for bordered Floer algebras derived from hypertoric categories.
problem Homological properties of bordered Floer algebras.
method Affine quasi hereditary property of equivariant hypertoric convolution algebras and computation of Ext groups.
result Existence of standard modules and isomorphism of Ext groups to bordered strands dg algebras.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using SU(2) singular instanton gauge theory, the paper constructs invariants and Morse chain complexes. result The constructions lead to a triad of groups and several concordance invariants.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…
This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with re…
Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.
problem Computing Floer homotopy types and eta invariants for Seifert 3-manifolds.
method Floer homology, Seiberg-Witten Floer homotopy type, adiabatic connections, spin^c-Dirac operators, eta invariants, orbifold pin^c-connections.
result Floer homotopy types are suspensions of S^0, and Seifert 3-manifolds are L-spaces.