We compare the homology groups of the chain complex of integral currents with compact support of a metric space with the singular Lipschitz homology and with ordinary singular homology. If satisfies certain cone inequalities all these homology theories coincide. On the other hand, for…
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48 results for “singular homology”
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that…
In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk (called a ration…
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
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 on homology groups of cDV singularity links, identifying their topology.
problem Identify the topology of links of cDV singularities of types and .
method Analyzing the second integral homology group of the links, using results from Smale and Thom-Sebastiani sums.
result The homology groups of the links are determined for cDV singularities of types and .
A Formula For the Homology Groups of the Milnor Fibres of an Isolated Real Hypersurface Singularitymath.AG
Formula calculates homology groups of Milnor fibres for real hypersurface singularities.
problem Calculating homology groups of Milnor fibres for real hypersurface singularities.
method Established a formula for homology groups of Milnor fibres relative to their boundaries.
result Formula provides a method to compute homology groups of Milnor fibres.
The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.
problem Defining and constructing a resolution cube for knot Floer homology of singular links in lens spaces.
method Defining grid homologies for singular links in lens spaces and using them to construct a resolution cube.
result A complete description of singular knot theory in lens spaces and a signed combinatorial resolution cube for knot Floer homology.
For an orbifold M we define a homology group, called t-singular homology group t-H_q(M), which depends not only on the topological structure of the underlying space of M, but also on the orbifold structure of M. We prove that it is a b-homotopy invariant of orbifolds. If M is a manifold, t-H_q(M) coincides with the usu…
We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
Cylindrical contact homology computed for links of simple singularities.
problem Computing cylindrical contact homology for links of simple singularities.
method Perturbing degenerate contact form on with an invariant Morse function, achieving nondegeneracy up to an action threshold. Recovers cylindrical contact homology via direct limit of action filtered homology groups.
result Ranks of cylindrical contact homology groups are given in terms of , demonstrating a form of the McKay correspondence.
We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
Injectivity proven for measure homology of certain wild spaces.
problem Injectivity of measure homology for mildly wild spaces.
method Proving injectivity of the canonical map from singular to measure homology.
result Injectivity of measure homology for certain mildly wild spaces.
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
Singular link Floer homologymath.GT
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler c…
We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…
Defines and calculates foliation homology from flows.
problem Homology of foliations defined by flows.
method Definition and calculation of foliation homology.
result Homology naturally associated with Seifert fibration.
Study symplectic cohomology of certain singularities using homological mirror symmetry.
problem Compute symplectic cohomology for specific singularities.
method Use homological mirror symmetry to compute symplectic cohomology.
result Suggests a new conjecture about the relationship between small resolutions and symplectic cohomology.
We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential compo…
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.
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
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…
Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …
Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.
problem Calculating instanton homology for specific braids.
method Using local coefficients and algebraic curves, calculates homology groups and their module structures.
result Equivalent to computing quantum cohomology of a moduli space of parabolic bundles.
In this paper we prove that the zeroth Milnor-Thurston homology group coincides with singular homology for Peano Continua. More- over, we show that the canonical homomorphism between these ho- mology theories may not be injective. However, it is proved that it is injective when a space has Borel path-components.
Paper proves conditions for rational homology complex projective planes with singularities.
problem Proving conditions for rational homology complex projective planes with singularities.
method Leveraging results from smooth 4-manifolds, including Donaldson diagonalization theorem and Heegaard Floer correction terms.
result Eliminates the possibility of a rational homology complex projective plane with four singularities and identifies families of singularities obstructed by smooth conditions.
While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities…
The paper introduces new knot invariants using singular instanton gauge theory.
problem Developing new knot invariants using singular instanton gauge theory.
method Using singular instanton gauge theory, the paper constructs invariants and Morse chain complexes.
result The constructions lead to a triad of groups and several concordance invariants.
Existence of non-trivial monopoles on 3-spheres proven.
problem Existence of non-trivial SU(2)-monopoles on 3-manifolds.
method Gluing construction to prove existence on rational homology 3-spheres.
result Existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities.
We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.
The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.
problem Understanding the homology of orbifolds.
method Constructing invariant and coinvariant Morse chain complexes for orbifolds.
result The homology of coinvariant Morse complexes computes the singular homology of the underlying space.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are a…
Constructs Morse homology for complex algebraic varieties.
problem Homology of vanishing cycles in complex algebraic varieties.
method Morse homology groups associated with a perturbed function on a compactified variety.
result Morse homology groups are isomorphic to the homology of vanishing cycles.
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
Extends Khovanov homology to surfaces with singularities.
problem Computing invariants for surfaces with singularities.
method Functorial extension of Khovanov homology to surfaces with double points.
result Induces a map between Khovanov homology groups of boundary links.
Paper categorifies Vassiliev skein relation for Khovanov homology.
problem Clarifying the relation between Vassiliev invariants and Khovanov homology.
method Developed a categorified version of Vassiliev skein relation on Khovanov homology.
result Khovanov homology's genus-one operation leads to a crossing change, enabling invariance under Reidemeister moves and extending to singular links.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex to a singular resolution of a knot . Manolescu conjectured that when is in braid position, the homology is isomorphic to the HOMFLY-PT homology of . Together with a naturality condition on t…
Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
Homologically area-minimizing surfaces mod have at worst codimension 2 singular sets asymptoticallymath.DG
New results show area-minimizing surfaces have fewer singularities than expected.
problem Understanding the singularities of area-minimizing surfaces in homology classes.
method Sharp regularity theorem for area-minimizing currents in finite coefficient homology.
result For large , area-minimizing mod currents are integral currents with a singular set of codimension at least 2.
Slicing degree of knotsmath.GT
The paper calculates the slicing degree of knots using advanced homology theories.
problem Determining the minimum slicing degree of knots.
method Rasmussen's s-invariant, knot Floer homology, and singular instanton homology.
result Computed slicing degrees for many small knots and some families of torus knots.
New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.
problem Finding new Sasaki-Einstein 7-manifolds and understanding their properties.
method Calculating homology groups of specific 7-manifolds using Thom-Sebastiani sums and quasi-regular metrics.
result 52 new Sasaki-Einstein rational homology 7-spheres and 124 new 2-connected 7-manifolds homeomorphic to were found.
Generalizes Lefschetz fibrations with rational homology disk smoothings.
problem Understanding rational homology disk smoothings of surface singularities.
method Introduces a genus to generic fibers of Lefschetz fibrations.
result Families of relations in mapping class groups represent smoothings.