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.
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.
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.
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.
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.
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.
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.
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.
Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.
problem Rank inequalities in Heegaard Floer homology.
method Using Hanselman-Rasmussen-Watson's bordered Floer homology, we extend their proof to rational homology solid tori.
result We provide rank inequalities for Heegaard Floer homology.
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.
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.
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.
New findings restrict Heegaard Floer homology for certain rational homology spheres.
problem The L-space conjecture or Heegaard Floer homology geography. method Verification of a stronger geography restriction for rational homology spheres.
result Heegaard Floer homology satisfies a stronger geography restriction for a wide class of rational homology spheres.
This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.
Survey on proof of homology isomorphism between two complex theories.
problem Proof of isomorphism between Heegaard Floer homology and embedded contact homology.
method Survey of proof methods from multiple papers.
result Established isomorphism between Heegaard Floer homology and embedded contact homology.
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.
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.
Study shows inequality in Floer homologies for 3-manifold covers.
problem Establishing an inequality between Heegaard Floer homologies of branched covers.
method Using Heegaard Floer homology, the study establishes an inequality for dimensions of homologies of 3-manifold covers.
result An inequality is established between the dimensions of Heegaard Floer homologies of branched covers.
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.
problem Proving a conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres.
method Using Zemke's isomorphism between lattice and Heegaard Floer homology, the paper proves Némethi's conjecture.
result The conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres is proven.
Proves lattice homology equals Heegaard Floer homology for certain 3-manifolds.
problem Equivalence of lattice homology and Heegaard Floer homology for specific 3-manifolds.
method Proves conjecture using plumbing of disk bundles over S^2.
result Lattice homology equals Heegaard Floer homology for 3-manifolds with specified properties.
Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
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.
We consider a stabilized version of hat Heegaard Floer homology of a 3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance pr…
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.
Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.
problem Computing Heegaard Floer homologies of Brieskorn spheres.
method Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong.
result Brieskorn spheres generate infinite rank summands in the homology cobordism group.
Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with ext…
We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to st…
Developed a real sutured Heegaard Floer theory.
problem Real sutured manifolds and their Floer homology.
method Real nice diagrams, combinatorial computation.
result New structural properties distinguishing from sutured Floer homology.
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic k-torsion in the context of ech, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heeg…
This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are pur…
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
Maps Heegaard Floer homology to Hecke algebras for surfaces.
problem Connecting Heegaard Floer homology with Hecke algebras.
method Constructs a map from Heegaard Floer homology to Hecke algebras.
result Shows the map is an isomorphism of algebras.
Decategorifies higher actions in Heegaard Floer homology.
problem Understanding higher actions in Heegaard Floer homology.
method Decategorification of bordered Heegaard Floer strands algebras and identification with actions on exterior powers of homology groups.
result Identifies decategorifications with certain actions on exterior powers of homology groups of surfaces.
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …
This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an a…
We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show…
New Heegaard Floer homology findings block chirally cosmetic surgeries.
problem Chirally cosmetic surgeries on knots and manifolds.
method Heegaard Floer homology, immersed curve formulations, and surgery formula.
result New obstructions to chirally cosmetic surgeries identified.
Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.
For any three-manifold presented as surgery on a framed link (L,Λ) in an integral homology sphere, Manolescu and Ozsváth construct a hypercube of chain complexes whose homology calculates the Heegaard Floer homology of Λ-framed surgery on Y. This carries a natural filtration that exists on any hypercube of chain comple…
Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.
problem Conditions for surgery on knots to produce non-separating spheres.
method Heegaard Floer homology
result Sufficient conditions for a knot to be unknotted.
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 compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.
Heegaard Floer homology connects to polynomial representations of Hecke algebras.
problem Understanding polynomial representations of double affine Hecke algebras.
method Using higher-dimensional Heegaard Floer homology and topological interpretations.
result Recovery of polynomial representations from Heegaard Floer homology.
A criterion for Whitney disks connects intersections in 3-manifold homology.
problem Existence of Whitney disks in Heegaard Floer homology.
method Use Nielsen theory to establish a criterion.
result Simple criterion for the existence of Whitney disks.