Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
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48 results for “homology equivalence”
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.
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
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.
Two definitions of set-theoretic Yang-Baxter homology are shown to be equivalent.
problem Equivalence of two Yang-Baxter homology definitions.
method Comparison of algebraic and graphic homology theories.
result The graphic homology is equivalent to the algebraic one.
Two Seiberg-Witten homologies are proven equivalent.
problem Equivalence of Seiberg-Witten homologies.
method Isomorphism proof between monopole Floer homology and Seiberg-Witten spectrum.
result Monopole Floer homology is isomorphic to Seiberg-Witten spectrum.
This paper is the sequel to "The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I" and is devoted to proving some of the technical parts of the HF=ECH isomorphism.
Proves Khovanov homology has no torsion for bipartite circle graphs.
problem Proving properties of Khovanov homology for bipartite circle graphs.
method Proved homotopy equivalence of independence complexes to wedges of spheres.
result Extreme Khovanov homology has no torsion.
For S a compact connected oriented surface, we consider homology cylinders over S: these are homology cobordisms with an extra homological triviality condition. When considered up to Y_2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian gro…
New local equivalence groups refine Rasmussen's s-invariant.
problem Refining Rasmussen's s-invariant for knot concordance.
method Introducing local equivalence groups combining Khovanov homologies.
result A refined s-invariant that determines triviality of knot images.
Introduces combinatorial tangle Floer homology and its strand diagram equivalent.
problem Defining and comparing tangle Floer homology in two equivalent ways.
method Presented strand diagrams and bordered grid diagrams, and discussed their equivalence.
result Explicit computations carried out to illustrate the equivalence of the two definitions.
New conditions prevent non-trivial relations in local equivalence group.
problem Understanding non-trivial relations in homology cobordism groups.
method Filtered instanton Floer homology and -invariants.
result Certain conditions prevent non-trivial relations in the local equivalence group.
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.
Homology 3-spheres are shown to be equivalent through specific twists.
problem Proving all homology 3-spheres are -equivalent.
method Exhibiting an element of acting trivially on the fourth nilpotent quotient of the fundamental group.
result All homology 3-spheres are -equivalent, providing an explicit example of a non-commutator element in .
Study -invariants for spherical 3-manifolds via -homology equivalences.
problem Computing -invariants for spherical 3-manifolds via -homology equivalences.
method Using Bott and Cattaneo's -invariants, defined by integrals over configuration spaces with local systems, and representation theory of finite groups.
result Computed upper bounds for dimensions of spaces spanned by -invariants and finite type invariants.
Study proves a space of manifolds is homology equivalent to a loop space.
problem Understanding the stable moduli space of certain high-dimensional manifolds.
method Incorporates surgery data via Lagrangian subspaces in a cobordism category.
result The stable moduli space is homology equivalent to an infinite loopspace.
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.
Study calculates Heegaard Floer homology for Seifert fibered 3-manifolds.
problem Computing Heegaard Floer homology for Seifert fibered 3-manifolds.
method Combinatorial models inspired by lattice homology.
result Connected Floer homology determines local equivalence class of -complex.
We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.
New invariants study Morse functions' equivalence classes in persistent homology.
problem Understanding equivalence classes of Morse functions on spheres for persistent homology.
method Graph-equivalent and height-equivalent Morse functions, with fundamental moves.
result Established new invariants to discern Morse functions more effectively.
The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.
problem Understanding Goeritz equivalence in lens spaces for genus two Heegaard splittings.
method Describes the Goeritz group action on the homology of the Heegaard surface and provides obstructions.
result Homology and homotopy obstructions for Goeritz equivalence of curves in the Heegaard surface.
Researchers debunk a generalized Property R conjecture for 2-component links.
problem The generalized Property R conjecture for 2-component links in .
method Disproved a further generalization of the GPRC using specific counterexamples.
result 2-component framed links in surging to a connected sum of homology 's are not handleslide equivalent to a split link.
Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology f…
New 4D examples show sphere equivalence doesn't imply isotopy.
problem Understanding when sphere equivalence implies topological isotopy in 4-manifolds.
method Constructing specific 4-manifolds with non-isotopic but equivalent 2-spheres.
result Gabai's 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds.
The paper studies properties of hyperbolic 3-manifolds and their equivalence.
problem Equivalence of left-orderable fundamental group, non-minimal Heegaard Floer homology, and taut foliation.
method New combinatorial criterion for taut foliations and improved methods for studying Heegaard Floer homology.
result Adds evidence supporting the conjecture about the equivalence of properties for rational homology 3-spheres.
The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plusmath.GT
Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded co…
We give a sufficient and necessary condition of the fundamental group homomorphism of a map between manifolds to induce homology equivalences. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.
New findings on 3-manifolds using Heegaard Floer theory.
problem Understanding equivalence classes of 3-manifolds.
method Heegaard Floer homology tools.
result Existence of simple balanced 3-manifolds not equivalent to .
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.
We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields.
We present three equivalent definitions of -equivariant symplectic homology. We show that, using rational coefficients, the positive part of -equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduc…
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.
Study stabilizes components of Galois cover moduli spaces.
problem Decide equivalence and stable equivalence of monodromy maps.
method Develop algebraic framework to study equivalence classes of monodromy maps.
result Recover a homological invariant that distinguishes equivalence classes.
We give a proof that the geometric K-homology theory for finite CW-complexes defined by Baum and Douglas is isomorphic to Kasparov's K-homology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariant K-homology theory.
Volume entropy semi-normmath.GT
Innovates volume entropy semi-norm, proving equivalence to simplicial volume.
problem Equivalence of volume entropy and simplicial volume in real homology.
method Introduces volume entropy semi-norm and proves its equivalence to simplicial volume.
result Equivalence of volume entropy semi-norm and simplicial volume in every dimension.
Coarse assembly mapsmath.KT
Coarse assembly maps generalize known results for K-homology.
problem Generalizing known results for K-homology to other coarse homology theories.
method Constructing coarse assembly maps as natural transformations between coarse homology theories.
result Explicit calculation of the domain of the coarse assembly map in terms of locally finite homology theory.
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.
Paper conjectures and proves simplicial complexes are homotopy equivalent to spheres, linking to knot theory.
problem Homotopy type of circle graphs complexes.
method Constructing and analyzing simplicial complexes from circle graphs.
result Simplicial complexes are homotopy equivalent to spheres in many cases.
We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.
Study shows knots in homology spheres can be topologically equivalent to those in .
problem Distinguishing knots in homology spheres from those in .
method Whitney tower filtration of concordance and solvable filtration.
result Every knot (or link) in a homology sphere is equivalent to a knot (or link) in modulo any term in the Whitney tower filtration.
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.
This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let and be 2-knots. Let and are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimen…
An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriatel…
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.
Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this B…
Homology from ternary algebras gives knot and surface invariants.
problem Creating knot and surface invariants using algebraic structures.
method Defining ternary algebra homology based on Reidemeister moves and quasigroups.
result Normalized homology yields invariants of knots and knotted surfaces.