For a closed symplectic manifold , a compatible almost complex structure , a 1-periodic time dependent symplectic vector field and a homotopy class of closed curves we define a Floer complex based on 1-periodic trajectories of in the homotopy class . We suppose that the closed 1-form …
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48 results for “Floer trajectories”
We study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic -family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of -dimensional (perturbed) -holomorphic maps to -dimensional gradient segments. We consider the …
With a view to constructing a Morse/Floer homology theory for CMC hypersurfaces, we prove a compactness result modulo broken trajectories for eternal mean curvature flows with forcing term in compact, hyperbolic manifolds.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
Defines Floer homology with DG coefficients for symplectic manifolds.
problem Computing Floer homology with DG coefficients for symplectic manifolds.
method Develops DG Floer toolset, defines spectral invariants, and proves Viterbo isomorphism theorem.
result Establishes almost existence of contractible periodic orbits on cotangent bundles.
We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…
This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I_F. (See math.DG/0111313 for part I). Having constructed I_F and outlined a proof of its invariance based on bifurcation analysis in part I, in this part we prove a series of gluing theorems to confirm the b…
This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.
problem Addressing the structure of path spaces on two-level manifolds.
method Introducing the notion of tameness and constructing charts on path spaces of two-level manifolds.
result Path spaces on tame two-level manifolds have the structure of a Hilbert manifold.
A 2-category categorifies complex Lagrangians in hyperkähler manifolds.
problem Categorify the Fukaya category of hyperkähler manifolds.
method Formalizes morphisms in a 2-category based on Fueter maps.
result Fueter maps correspond to complex gradient trajectories in cotangent bundles.
Develops Morse homology with DG coefficients for manifolds and spaces.
problem Homology with DG coefficients for manifolds and spaces.
method Derived local systems, DG modules, twisting cocycles, Morse trajectories.
result Isomorphic to DG Tor and Ext functors, recovers homology of total space of fibrations.
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with -cables and -cables.
Proves properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Knot Floer homology matches fixed point Floer for fibred knots.
problem Understanding the algebraic structure of knot Floer homology.
method Proved isomorphism between Knot Floer and fixed point Floer homologies.
result Knot Floer homology of fibred knots matches fixed point 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.
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.
Introduces Floer lasagna modules using link Floer homology.
problem No specific problem stated; focuses on new mathematical concept.
method Inspired by skein lasagna module, uses link Floer homology.
result Computes Floer lasagna modules for specific 4-manifolds.
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 Lagrangian Floer theory in smooth divisor complements.
problem Intersection Floer theory of Lagrangians in smooth divisor complements.
method Complete construction of Floer homology.
result Floer homology for Lagrangians in smooth divisor complements constructed.
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
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.
Combinatorial definition of bordered Floer theory for torus-boundary manifolds.
problem Defining Floer homology for manifolds with torus boundaries.
method Combinatorial definition with coefficients.
result Recovery of combinatorial Heegaard Floer homology.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
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.
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.
New colored knot Floer homology defined using infinite full twists.
problem Defining a new homology theory for knots.
method Defining colored knot Floer homology through colimit of link Floer homology with infinite full twists.
result Colored knot Floer homology is a module over the colored knot Floer homology of the unknot.
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …
Real Heegaard Floer Homologymath.GT
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.
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.
Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for -equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to S…
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.
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.
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 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.
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.
Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…
A formula for bordered Floer homology of concordances and satellites
problem Computing knot Floer homology for concordances and satellites
method Combinatorial method for bordered Floer homology
result Computation of knot Floer homology cobordism map
The paper shows infinitely many components in Floer Hessians space.
problem Understanding the structure of Floer Hessians.
method Proving the existence of infinitely many connected components.
result Proves infinitely many connected components in Floer Hessians space.
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.
Survey of bordered Floer homology for 3-manifolds with boundary.
problem Computing invariants for 3-manifolds with boundary.
method Extends Heegaard Floer homology to include boundary.
result Powerful computational tools for 3-manifolds with boundary.
Floer homology linked to Milnor fibers for certain singularities.
problem Understanding Floer homology of singularities using Milnor fibers.
method Combining Floer theory with Milnor fibers, using combinatorial formulas.
result Equality of Casson invariants in Donaldson and Seiberg-Witten theories.
Floer homology detects right-veering monodromy in fibered knots.
problem Detecting right-veering monodromy in fibered knots.
method Using Heegaard Floer homology and symplectic Floer homology.
result Knot Floer homology characterizes tight contact structures.
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…
Formula for satellite operators using knot Floer homology.
problem Computing knot Floer homology for satellite knots.
method Using Heegaard Floer Dehn surgery formulas and formal knot Floer complexes.
result Formulas to compute knot Floer homology for satellite knots.
Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
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.
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.