The study explores properties of homologically locally connected spaces and their connections to other topological concepts.
problem Characterizing and understanding homologically locally connected spaces.
method Investigates properties and characterizations of homologically UVn-maps and lcGn-spaces, comparing them to existing concepts. result Identifies similarities and parallels between homologically locally connected spaces and other topological concepts.
We extend the cobordism based categorification of the virtual Jones polynomial to virtual tangles. This extension is combinatorial and has semi-local properties. We use the semi-local property to prove an applications, i.e. we give a discussion of Lee's degeneration of virtual homology.
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.
The paper confirms properties of homogeneous ANR compacta and their homological similarities.
problem Properties of homogeneous ANR compacta and their homological similarities.
method Analyzes local homological properties and cyclicity of homogeneous ANR compacta.
result Homogeneous ANR compacta have similar local homological properties to closed balls and their boundaries.
Enhances knot Floer homology with algebraic representation theory.
problem Compatibility between summands in bordered knot Floer homology.
method Categorifies intertwining property of higher representations.
result New algebraic reformulation of compatibility property.
We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characte…
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae, 71 (1983) 77-129] in their proof of topological invariance of intersection homology, homology stratifications do not appe…
The study explores how 3-manifolds embed locally flatly in S4.
problem Embedding homology 3-spheres in S4 locally flatly. method Survey of known results, focusing on Seifert fibred 3-manifolds and using 4-dimension surgery.
result Characterization of embeddings by properties of complementary regions.
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.
The paper extends local calibration pairs to global ones and finds mass-minimizing submanifolds.
problem Extending local calibration pairs to global ones in various Riemannian manifolds.
method Analyzing mass-minimizing properties and using conformal classes.
result Homologically mass-minimizing submanifolds exist in some Riemannian manifolds that cannot be calibrated by smooth calibrations.
Paper explores topological invariance in torsion-sensitive intersection homology.
problem Torsion-sensitive intersection homology's topological invariance.
method Analogous to classical intersection homology, investigates invariance of sheaf complexes.
result Provides torsion-sensitive versions of existing invariance theorems and new results.
In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local co…
Differential refinement of homology satisfies Poincaré duality.
problem Constructing a refined homology theory that satisfies Poincaré duality.
method Defining differential cap product and constructing differential refinement of singular homology.
result Essential uniqueness of differential homology and cap product proven.
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
Study properties of algebraic threefolds with specific singularities.
problem Characterize and classify algebraic threefolds with certain singularities.
method Use topological invariants, rational homology, Poincaré duality, and Lie algebras.
result Relate topological invariants to Lie algebras and representations.
Link Floer homology is split into snake complexes and local systems.
problem Classifying link Floer complexes over specific rings.
method Classifying isomorphism and chain homotopy equivalence classes of free chain complexes over a specific ring, then applying these results to link Floer complexes.
result Link Floer complexes split uniquely into snake complexes and local systems.
Defines a local Heegaard Floer theory for tangles.
problem Develops a Heegaard Floer theory for tangles.
method Defines a local version of Heegaard Floer homology for tangles, proving glueing theorems and skein relations.
result Shows links related by mutation about a specific tangle have the same δ-graded link Floer homology.
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.
Study on how non-reversible diffusion processes affect homology on manifolds.
problem Understanding the asymptotic behavior of random homology in diffusion processes.
method Investigation of asymptotic properties of random homology associated with stochastic diffusion processes on compact Riemannian manifolds.
result For quadratic rate, manifold is a locally trivial fiber bundle over a flat torus with minimal fibers.
Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.
problem Classifying spaces as almost homology n-manifolds.
method Providing a necessary and sufficient condition for locally compact homogeneous ANR-spaces or strongly locally homogeneous ANR-spaces to be almost homology n-manifolds.
result Spaces are classified as almost homology n-manifolds based on their homology groups.
The paper studies geometric properties of geodesic spaces using Rips and Čech filtrations.
problem Understanding geometric properties of geodesic spaces.
method Applying fundamental group and homology groups to Rips or Čech filtrations.
result Rips critical points correspond to circles of specific lengths and persistence encodes space properties.
Study covers of surfaces, showing types and properties.
problem Characterize homeomorphism types of surface covers.
method Analyzing homeomorphism types of covers of surfaces, identifying universal covers and homology covers.
result Identify four possible homeomorphism types for certain covers of surfaces.
Investigates local indicability of groups with circle homology presentations.
problem Conditions for local indicability in groups with circle homology presentations.
method Generalizes results for two-relator presentations to circle homology presentations.
result Extends results on local indicability to LOT groups and non-cycle-free Adian presentations.
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
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.
The paper studies twisted Morse homology and cohomology on manifolds.
problem Computing homology and cohomology with local coefficients on manifolds.
method Morse theory, CW-complexes, de Rham cohomology, Lichnerowicz cohomology.
result Isomorphisms between different cohomology theories.
Local biquandles link link coloring to tribracket theory.
problem Link coloring and tribracket theory.
method Introduced local biquandles and defined their cohomology.
result Local biquandle cohomology is isomorphic to Niebrzydowski's tribracket cohomology.
New instanton homology for webs counts Tait colorings.
problem Counting Tait colorings of complex webs.
method Introducing a local system of coefficients to deform instanton homology.
result Rank of deformed instanton homology equals number of Tait colorings.
Collects properties of Seifert homology spheres for concordance studies.
problem Understanding concordance properties of fibers in Seifert homology spheres.
method Using Dehn surgery along Seifert fibers.
result Includes various properties in one place for concordance studies.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
Grid homology properties for MOY graphs studied.
problem Defining and studying properties of grid homology for MOY graphs.
method Defined grid homology from Harvey and O'Donnol's work. Studied properties using oriented skein relation, edge contraction, and parallel edge unification.
result Properties of grid homology for MOY graphs were studied and defined.
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the fini…
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.
Shadow biquandles and local biquandles have similar homology and invariants.
problem Comparing homology and invariants of shadow biquandles and local biquandles.
method Defined local biquandle structure on a shadow biquandle and showed isomorphic (co)homology groups and invariant equivalence.
result Homology and invariants of shadow biquandles and local biquandles are equivalent.
The abstract introduces a sequence of moves between two types of Heegaard diagrams for knot Floer homology.
problem Using different Heegaard diagrams for knot Floer homology.
method Explicit sequence of Heegaard moves connecting Kauffman-states and planar diagrams.
result Local moves can be used to transform between global Heegaard diagrams.
Every open manifold L of dimension greater than one has complete Riemannian metrics g with bounded geometry such that (L,g) is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the local geometry of (L,g) suffice to make it quasi-isometric to a leaf of such a foli…
Proves transversality for local Morse homology with symmetries.
problem Defining local Morse chain complexes with finite cyclic group symmetry.
method Special regularized distance functions and inductive perturbation process.
result Global existence theorem for symmetric Morse-Smale pairs.
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.
The paper reveals a property of chromatic homology for complete graphs.
problem Understanding the chromatic homology of complete graphs.
method Introduced a combinatorial description of enhanced states and used it to analyze the homology.
result Showed a splitting property of the chromatic homology for certain graphs.
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.
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 rs-invariants. result Certain conditions prevent non-trivial relations in the local equivalence group.
Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.
problem Counting flat cycles in congruence covers of symmetric spaces.
method Lower bound calculation for immersed compact flat manifolds.
result Lower bounds on the contribution of flat cycles to homology.
Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n≥3.
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 ι-complex. 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.
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres