Short note proves Lie algebroid deformation cohomology preservation under fibration conditions.
problem Understanding deformation cohomology of Lie algebroids under fibration conditions.
method Proves preservation of deformation cohomology up to degree k for Lie algebroids under certain fibration conditions.
result Deformation cohomology of Lie algebroids is preserved up to degree k under specified fibration conditions.
We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H_{k+1}(N;Z_2) on the set of embeddings. (For k\ne1 classification results w…
Using the concept of s-formality we are able to extend the bounds of a Theorem of Miller and show that a compact k-connected 4k+3- or 4k+4-manifold with b_{k+1}=1 is formal. We study k connected n-manifolds, n= 4k+3, 4k+4, with a hard Lefschetz-like property and prove that in this case if b_{k+1}=2, then the manifold i…
Reduces k-symplectic field theory equations and discusses reconstruction methods.
problem Reduction and reconstruction of k-symplectic field theory equations.
method Explicit coordinate expressions of reduced equations, interpretation of integrability conditions, introduction of k-connection.
result Invariant Lagrangian defines a mechanical k-connection for reconstruction.
Let X be a differentiable manifold endowed with a transitive action α:A×X⟶X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equ…
We show that for every symmetric space G/K of compact type with K connected, the K-action on G/K by left translations is equivariantly formal.
Flat connections derived from Poisson brackets on loop spaces.
problem Understanding the structure of Poisson brackets on loop spaces.
method Defined connections by explicit linear combinations of standard connections associated with the Poisson bracket.
result Connections are shown to be flat.
The paper extends the formal manifold theorem to higher dimensions and characterizes A∞-minimal models for certain differential graded algebras.
problem Characterizing formal differential graded algebras and their A∞-minimal models. method Expanding the formal manifold theorem and proving properties of A∞-minimal models for specific cases. result The de Rham complex of certain differential graded algebras has A∞-minimal models with specific non-trivial terms. An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…
The paper proves eigenvalues are simple for specific operators on bundles.
problem Eigenvalue simplicity for connection Laplacian and G-simplicity on bundles. method Analyzes connections on vector bundles and principal bundles, proving eigenvalue simplicity for a residual set of connections.
result Eigenvalues of the connection Laplacian and Laplace-Beltrami operator are simple for specified conditions.
New groups Γnk connect braids and 3-manifolds.
problem Understanding relationships between braids and 3-manifolds.
method Introducing and studying groups Γnk. result Groups Γn4 lead to new relationships between braids and manifolds. New findings on metric spaces with finite Nagata dimension.
problem Understanding isoperimetric properties in subsets of metric spaces.
method Analyzing quasiconvex subsets with finite Nagata dimension and applying isoperimetric inequalities.
result Quasiconvex subsets of metric spaces with finite Nagata dimension are isoperimetrically undistorted up to a certain dimension.
We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near …
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in se…
The abstract discusses embedding manifolds in open books and contact structures.
problem Embedding manifolds in open books and contact structures.
method Using open book embeddings and contact structures, the abstract proves embedding theorems for various manifolds.
result Closed manifolds can be embedded in open books and contact structures.
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…
The paper finds solutions to the Yamabe equation with low energy and multiple nodal sets.
problem Finding solutions to the Yamabe equation with specific nodal structures.
method Reduction to a singular ODE and solving a double shooting problem.
result Existence of solutions with exactly k zeroes, yielding k connected isoparametric hypersurfaces as nodal sets.
This paper studies the interplay between the N=2 gauge theories in three and four dimensions that have a geometric description in terms of twisted compactification of the six-dimensional (2,0) SCFT. Our main goal is to construct the three-dimensional domain walls associated to any three-dimensional cobordism. We find t…
The paper simplifies smooth maps to spheres and planes, showing homotopy and embedding properties.
problem Constructing explicit deformations of smooth maps to spheres and planes.
method Explicit constructions and homotopy arguments.
result Smooth maps are homotopic to stable maps with limited singularities.
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.
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
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.
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.
Unified framework for proving functoriality of various link homology theories.
problem Functoriality and Reidemeister invariance of link homology theories.
method Unified framework leveraging Khovanov homology and various link homology theories.
result Stronger functoriality results by avoiding spectral sequences.
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.
Khovanov and chromatic homologies show similar patterns, improving chromatic bounds and computing Jones polynomial.
problem Understanding similarities between Khovanov and chromatic homologies.
method Analyzing isomorphism and improving bounds using explicit formulas.
result Improved bounds for chromatic homology and explicit formula for chromatic homology rank.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
The paper introduces representation homology of topological spaces and its connections to other homology theories.
problem Understanding the algebraic structure of topological spaces through representation homology.
method Developed a geometric relation between representation homology and higher Hochschild homology, constructed maps and spectral sequences, and computed explicit examples.
result Representation homology of the suspension of a space is isomorphic to its higher Hochschild homology.
New operations on Khovanov homology refine knot invariants.
problem Understanding finer knot invariants through homological operations.
method Developed an algebra of homological operations on Khovanov homology and lifted it to integral Khovanov homology.
result Conjectured infinite algebras of homological operations and provided evidence.
The paper examines lattice homology invariants of Seifert homology spheres.
problem Understanding homology cobordism invariants for Seifert fibered integral homology 3-spheres.
method Utilizes lattice homology and Heegaard Floer homology to study invariants.
result Reproves and extends the invariance of Seifert homology spheres' d-invariants and maximal monotone subroots. Detects figure-eight knot using Khovanov homology.
problem Detecting the figure-eight knot.
method Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology.
result Reduced Khovanov homology (over Q) detects the figure-eight knot.
Study improves Heegaard Floer homology relations for knots in homology spheres.
problem Improving relations in Heegaard Floer homology for knots in homology spheres.
method Proved inequality for d-invariants, used reduced Floer homology rank relations.
result Degree one maps between aspherical Seifert homology spheres are homotopic to homeomorphisms if Heegaard Floer homologies are isomorphic.
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.
Proves a rank inequality between Khovanov and knot Floer homologies.
problem Rank inequality between Khovanov and knot Floer homologies for knots.
method Oriented cube of resolutions construction for knot Floer homology.
result Proves Rasmussen's conjecture about Khovanov and knot Floer ranks.
Corrects a lemma in a 2009 paper about contact homology and Floer homology.
problem An error in a lemma about contact homology and Floer homology.
method None, as it is a correction of an existing lemma.
result Corrects an error in a previously published lemma.
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.
Khovanov-Floer theories are shown to be invariant under mutation.
problem Invariance of Khovanov-Floer theories under Conway mutation.
method Spectral sequences from Khovanov homology and proofs of conjectures.
result Strong Khovanov-Floer theories are mutation-invariant.
Study shows infinite-rank summand in homology concordance group of knots.
problem Homology concordance of knots in integer homology three-spheres.
method Knot Floer homology to construct homology concordance homomorphisms.
result Homology concordance group modulo knots from S^3 contains an infinite-rank summand.
Lower bounds on ribbon distance using Bar-Natan and α-Homology.
problem Calculating the minimum number of ribbon operations to unknot a knot.
method Bar-Natan Homology and α-Homology approaches.
result Lower bounds on ribbon distance via both Bar-Natan and α-Homology.
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.
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 (n,nm)-cables and (2,2m)-cables. Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
We compare the homology groups HnIC(X) of the chain complex of integral currents with compact support of a metric space X with the singular Lipschitz homology HnL(X) and with ordinary singular homology. If X satisfies certain cone inequalities all these homology theories coincide. On the other hand, for…