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.
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.
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.
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.
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.
New invariants improve Heegaard Floer slice genus and clasp number bounds.
problem Improving bounds for knot concordance.
method Using knot Floer homology and involutive correction terms.
result Improved slice genus and clasp number bounds proved.
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involu…
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.
The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.
problem Calculating numerical invariants for specific 3-manifolds.
method Involutive Heegaard Floer homology techniques and spin filling constraints.
result Established new constraints and obstructions for 3-manifolds.
Proves spectral sequence for real Heegaard Floer homology.
problem Real Heegaard Floer homology and its localization.
method Proves existence of a localization spectral sequence.
result Existence of spectral sequence for real Heegaard Floer hat variant.
Real bordered Floer homology computes 3-manifolds with involution.
problem Computing real Heegaard Floer homology for 3-manifolds with involution.
method Using bordered Heegaard Floer algebra modules, a practical algorithm is given.
result Computes real Heegaard Floer homology for real 3-manifolds with connected fixed set.
New invariants for 4-manifolds obstruct certain surface pairs.
problem Obstructing the existence of specific surface pairs in 4-manifolds.
method Defining new mixed invariants using involutive Heegaard Floer homology.
result The new invariants obstruct the existence of certain surface pairs.
Study homology cobordism and plumbed manifolds using Heegaard Floer homology.
problem Understanding homology cobordism and local equivalence between plumbed manifolds.
method Use involutive Heegaard Floer homology to analyze linear combinations of almost-rational plumbed three-manifolds.
result Proves that certain plumbed manifolds are not torsion in the homology cobordism group.
Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z 4 \mathbb{Z}_4 Z 4 -equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d ‾ \underline{d} d and d ˉ \bar{d} d ˉ …
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.
Introduces integer-valued Heegaard Floer theory with canonical orientations.
problem Defining and proving properties of Heegaard Floer homology over integers.
method Using canonical orientations from coupled Spin structures, proving naturality and surgery exact triangle.
result Established integer-valued Heegaard Floer theory and proved its properties.
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.
We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with d ‾ ( Y ) ≠ d ( Y ) ≠ d ‾ ( Y ) \underline{d}(Y) \neq d(Y) \neq \overline{d}(Y) d ( Y ) = d ( Y ) = d ( Y ) . We also construct a homomorphism from the three-dimensional homolo…
Real Heegaard Floer homology gets a new grading for certain 3-manifolds.
problem Real Heegaard Floer homology groups get an absolute Z/2 grading under specific conditions.
method Analyzes real Heegaard Floer homology groups with an involution and nullhomologous fixed points.
result Defines a new invariant of knots equal to the Alexander polynomial evaluated at i.
New knot invariants from Floer theory help bound knot three-genus.
problem Bounding the three-genus of knots.
method Involutive Heegaard Floer knot theory.
result Two new concordance invariants defined.
New homomorphism proven using immersed curves on disks.
problem Existence of homomorphisms and summands in homology groups.
method Approximating involutive Heegaard Floer complexes with immersed curves on the twice punctured disk.
result Existence of a new homomorphism and a Z ∞ \mathbb{Z}^{\infty} Z ∞ summand proven. New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
We show that Bordered Heegaard Floer invariant C F D ^ \widehat{CFD} C F D of a knot complement in S 3 S^3 S 3 is invariant under the elliptic involution on its boundary.
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.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
New surgery exact triangles in Heegaard Floer homology for rational slopes.
problem Constructing new surgery exact triangles in Heegaard Floer homology.
method Combining combinatorial triangle and quadrilateral counting in genus 1 Heegaard diagrams.
result Solving the combinatorial problem for rational slopes, including tricky cases.
Projective naturality proved for Heegaard Floer homology.
problem Proving naturality of Heegaard Floer invariants under diffeomorphisms.
method Showed Heegaard Floer invariants yield functors to transitive systems in a projectivized category of Z [ U ] \mathbb{Z}[U] Z [ U ] -modules. result Established Heegaard Floer invariants as functors to transitive systems in a projectivized category.
Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Fl…
In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms d ‾ , d ‾ \underline{d},\overline{d} d , d , are related to the P i n ( 2 ) \mathrm{Pin}(2) Pin ( 2 ) -equivariant Floer homology S W F H ∗ G \mathit{SWFH}^G_* SWFH ∗ G . We show that the only interesting correction terms of a $\mathrm{Pin}…
Study shows infinite-rank summand in homology cobordism group.
problem Identifying infinite-rank summands in homology cobordism groups.
method Introduced an algebraic variant of the involutive Heegaard Floer package.
result Demonstrated an infinite-rank summand in the homology cobordism group.
We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.
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.
Lecture notes on Heegaard Floer homology for 3-manifolds and knots.
problem Understanding Heegaard Floer homology for 3-manifolds and knots.
method Explains Heegaard diagrams, Heegaard Floer homology, and knot Floer homology.
result Relationship between knot and 3-manifold invariants.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Heegaard Floer homology study confirms composition maps match up to homotopy.
problem Verifying consistency of composition maps in Heegaard Floer homology.
method Using Auroux and Zemke's results, proving agreement up to homotopy.
result Proves consistency of composition maps in Heegaard Floer homology.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
Proves a formula in Heegaard Floer homology using combinatorial methods.
problem Proving Lipshitz's Maslov index formula in Heegaard Floer homology.
method Combinatorial proof via Heegaard diagrams.
result Validated Lipshitz's Maslov index formula in Heegaard Floer homology.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
Heegaard Floer homology duality applied to 4D mapping tori.
problem Computing Heegaard Floer invariants of 4D mapping tori.
method Applied Heegaard Floer homology and duality theory.
result Computed invariants of 4D mapping tori using Lefschetz numbers.
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
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 \mathbf{Z} Z -valued framed real Seiberg--Witten invariants for 4-manifolds with involutions. Formula for Heegaard Floer multicurves of double tangles from knot complements.
problem Computing Heegaard Floer multicurve invariants of double tangles.
method Using a simple formula derived from knot complements, comparing with Khovanov homology.
result New characterisation of L-space knots and first example of thin knot Floer homology.
Lecture notes on Heegaard Floer homology for beginners.
problem Introducing Heegaard Floer homology to newcomers.
method Illustrates algebraic structures via grid homology, then defines Heegaard Floer homology geometrically.
result Defines and describes key properties of Heegaard Floer homology.
Paper classifies Heegaard Floer minimal knots in sutured manifolds.
problem Classifying minimal knots in sutured manifolds.
method Uses Heegaard Floer homology.
result Agrees with instanton Floer homology classification when applicable.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
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.
New Heegaard Floer homology for orbifolds with cyclic singularities.
problem Defining Heegaard Floer homology for orbifolds with arbitrary cyclic singularities.
method Generalizing recent work by defining Heegaard Floer homology for orbifolds with arbitrary cyclic singularities.
result Defined a new Heegaard Floer homology for orbifolds with arbitrary cyclic singularities.