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.
The paper develops a new Floer theory for 3-manifolds with involutions.
problem Constructing a Floer theory for 3-manifolds with symmetries.
method Develops a framed real monopole Floer homology for 3-manifolds with involutions and provides definitions for invariants.
result Defines Z-valued framed real Seiberg--Witten invariants for 4-manifolds with involutions. New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
Real Heegaard Floer Homology extends Li's real monopole Floer homology.
problem Computing invariants of three-manifolds with symmetries.
method Real version of Lagrangian Floer homology, combinatorial computation.
result Euler characteristic of real Heegaard Floer homology is a knot invariant.
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).
Formula connects Lefschetz number to Frøyshov invariant and Murasugi signature.
problem Calculating the Lefschetz number for a specific map in Floer homology.
method Skein-theoretic argument and exact triangle in monopole Floer homology.
result Formula for Lefschetz number in terms of Murasugi signature and Frøyshov invariants.
Study h-cobordisms of complexity 2 in 5D, finding obstructions and examples.
problem Understanding h-cobordisms of complexity 2 in 5D. method Compute monopole Floer homology and action of twisting involution.
result Obtained obstructions and constructed examples of high complexity.
Study contact structures and sutured monopole Floer homology, computing and deriving formulas.
problem Understanding the relationship between contact structures and sutured monopole Floer homology.
method Exploring contact elements under Floer excisions, computing sutured monopole Floer homology, and deriving formulas.
result Obtained exact triangles and connected sum formulas for sutured monopole Floer homology.
Paper constructs maps for sutured monopole Floer homology.
problem None explicitly stated in the abstract.
method Constructs gluing and cobordism maps for sutured monopole Floer homology.
result Developed mathematical tools for sutured monopole Floer homology.
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
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.
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.
Paper generalizes sutured Floer homologies with new algorithms and polytopes.
problem Computing and understanding sutured Floer homologies for complex manifolds.
method Grading shifting property, algorithms, polytopes, canonical decompositions.
result Established new bounds and detection results for sutured Floer homologies.
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.
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.
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.
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
Study SOLV geometry using monopole Floer homology.
problem Prove SOLV rational homology sphere Y is an L-space.
method Apply Fourier analysis on solvable groups to show irreducible solutions do not exist for certain metrics.
result Y is an L-space geometrically proven.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
Lecture notes on monopole Floer homology, including differential geometry and correction terms.
problem Defining and understanding monopole Floer homology.
method Explains differential geometry, Morse theory, and four-dimensional theory connections.
result Sketches the relation to Manolescu's disproof of the Triangulation Conjecture.
Develops invariants for webs and foams using Seiberg-Witten theory.
problem Computing invariants for complex geometric structures.
method Monopole Floer homology with orbifold singularities.
result New invariants for webs and foams.
Study Seiberg-Witten theory on manifolds with codimension-3 foliations.
problem Analyzing Seiberg-Witten theory on manifolds with codimension-3 Riemannian foliations.
method Construct basic Seiberg-Witten invariant and monopole Floer homologies for each transverse spin structure.
result Monopole Floer homologies are independent of the bundle-like metric and generic perturbation.
Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
Involutive bordered Floer homology computes 3-manifold invariants.
problem Computing involutive Heegaard Floer homology for 3-manifolds.
method Bordering technique to extend involutive HF-hat, algorithmic approach, mapping class group action computation.
result Involutive HF-hat satisfies a surgery exact triangle, computed for many knots.
Third paper in series defines monopole Floer homology and gluing theorem for 3-manifolds.
problem Define and prove gluing theorem for monopole Floer homology.
method Define monopole Floer homology, establish gluing theorem for disconnected boundary.
result Floer homology detects Thurston norm and fiberness of 3-manifolds.
Unified tau invariants in instanton and monopole Floer theories.
problem Unified tau invariants in instanton and monopole Floer theories.
method Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
result Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
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. We prove the existence of an exact triangle for the Pin(2)-monopole Floer homology groups of three manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this tr…
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.
New spectral sequence connects Khovanov homology to real monopole Floer homology.
problem Computing real monopole Floer homology for links.
method Defined a spectral sequence from reduced Khovanov homology of the mirror link.
result Spectral sequence abuts to real monopole Floer homology of a link.
We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0.
Proves surgery exact triangle for monopole Floer homology over integers.
problem Proving the surgery exact triangle for monopole Floer homology over integer coefficients.
method Modification of Kronheimer--Mrowka's local system and adaptation of Freeman's computation.
result Obtains a spectral sequence over integer coefficients for an oriented link in S3. The paper uses Heegaard Floer homology to study homology cobordism groups.
problem Understanding the structure of homology cobordism groups.
method Defined an invariant using Heegaard Floer homology, analogous to Stoffregen's connected Seiberg-Witten Floer homology.
result Computed involutive correction terms for certain three-manifolds.
Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homolo…
Real Seiberg-Witten and monopole Floer homologies are equivalent for certain 3-manifolds.
problem Equivalence of two Floer homologies for real structures.
method Direct isomorphism proof between real Seiberg-Witten and monopole Floer homologies.
result Real Seiberg-Witten and monopole Floer homologies are isomorphic.
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.
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.
problem Understanding the homology cobordism group of 3-manifolds formed by knots and their mirrors.
method Using involutive Floer homology and gauge theory.
result Locally trivial involutive Floer homology on certain splices of knots and their mirrors.
Study Heegaard Floer homology of plumbed 3-manifolds, proving infinite-rank subgroup existence.
problem Computing Heegaard Floer homology for specific 3-manifolds.
method Involutive Heegaard Floer homology calculations.
result Existence of an infinite-rank subgroup in the homology cobordism group.
Formula connects Heegaard Floer homology terms for 3-manifolds.
problem Formula for involutive Heegaard Floer homology terms.
method Connected sum formula for involutive Heegaard Floer homology.
result Example of 3-manifold with non-standard correction terms.
Defines monopole Floer homology for 3-manifolds with toroidal boundaries.
problem Calculating invariants of 3-manifolds with toroidal boundaries.
method Develops monopole Floer homology using 3+1 topological quantum field theory.
result Euler characteristic recovers Milnor-Turaev torsion invariant.
Proves equivalence of contact invariants in two Floer homology theories.
problem Identifying contact invariants in different Floer homology theories.
method Proves isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology.
result Identifies contact class with Legendrian invariants in knot Floer homology.
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…
Monopole Floer homology connects 3-manifold invariants to Riemann surface geometry.
problem Relating 3-manifold invariants to Riemann surface geometry.
method Connecting monopole Floer homology to theta characteristics and spin structures.
result Monopole Floer homology groups of mapping tori are determined by automorphisms of Riemann surfaces.
Smith inequality proven for 3-manifold covers in Floer homology.
problem Understanding regular covers of 3-manifolds in Floer homology.
method Monopole Floer homology and Heegaard Floer correspondence.
result 3-manifolds with certain cover properties are L-spaces.
Corrected mistakes in knot Floer homology formulas for spliced knot complements.
problem Incorrect assumption about knot Floer homology involution.
method Considered the contribution from a possibly non-trivial involution.
result Corrected splicing formulas in knot Floer homology.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.