New cohomology defined for Lie algebroids by odd cocycles.
problem Defining a new cohomology for Lie algebroids.
method Defining twisted cohomology by an odd cocycle in Lie algebroid cohomology.
result Twisted cohomology only depends on the Lie algebroid cohomology class of the odd cocycle.
Researchers twist Deligne cohomology for the first time.
problem No new problem introduced; existing Deligne cohomology is already versatile.
method Explicitly twisted Deligne cohomology by taking degree one twists of integral and de Rham cohomology.
result New properties of twisted Deligne cohomology are presented and illustrated.
Extends cohomology theory for infinite volume transformation groups.
problem Generalizing cohomology for infinite volume transformation groups.
method Introduces norm-controlled cohomology as a generalization of bounded cohomology.
result Establishes norm-controlled cohomology for infinite volume transformation groups.
New cohomology theory for Lie 2-algebras extends classical theory.
problem Classical cohomology theory limitations for Lie 2-algebras.
method Introduced a new cohomology theory for Lie 2-algebras.
result Second cohomology group classifies extensions of Lie 2-algebras.
New metric space cohomology relates to Riemannian manifold cohomology.
problem Understanding cohomology structures on Riemannian and contact manifolds.
method Relating Lq,p-cohomology to ℓq,p-cohomology and proving quasi-isometry invariance. result Quasi-isometry invariance and multiplicative structure of Lq,p-cohomology. Generics extended to new cohomologies.
problem Extending classical genera to new cohomologies.
method Constructing generalised characteristic classes for bordism cohomologies.
result Natural extension of classical genera to new cohomologies.
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
The article examines twisted cohomologies on algebraic and analytic varieties.
problem Understanding and comparing twisted cohomologies on algebraic and analytic varieties.
method Comparison and definition of twisting parameters in both categories, algebraic and analytic.
result Reviewed isomorphisms of twisted cohomologies for cohomologous twisting parameters.
Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.
problem Understanding cohomology of hemistrict Lie 2-algebras.
method Functorial construction and isomorphism proof of cohomology.
result Cohomology of hemistrict Lie 2-algebras is isomorphic to Chevalley-Eilenberg cohomology.
New cohomology theories for heaps and ternary operations linked to group cohomology.
problem Defining and studying cohomology theories for heaps and ternary operations.
method Introduced para-associative and heap cohomology theories, and ternary self-distributive cohomology with abelian heap coefficients.
result Heap cohomology is related to group cohomology via a long exact sequence, and injects into ternary self-distributive cohomology.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…
De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. New proof of blow-up formula for Morse-Novikov cohomology.
problem Blow-up formula for Morse-Novikov cohomology.
method Introducing relative Morse-Novikov cohomology and using sheaf cohomology.
result Explicit isomorphism in relative Morse-Novikov cohomology.
Survey on equivariant cohomology of Lie group actions.
problem Understanding Lie group actions on manifolds.
method Focus on equivariant formality and de Rham theory.
result Applications to ordinary cohomology and fixed points.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
Rham cohomology of Lie groupoids over triangulated manifolds is isomorphic to piecewise cohomology.
problem Establishing Rham cohomology for Lie groupoids over triangulated manifolds.
method Using isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, proving Rham cohomology is isomorphic to piecewise Rham cohomology.
result Piecewise de Rham cohomology of a Lie groupoid does not depend on the triangulation of the base.
Compute local cohomology of vector fields on manifolds.
problem Understanding cohomology of vector fields on manifolds and complex manifolds.
method Compute local cohomology, use descent for cocycles.
result Explicit representatives for cocycles constructed.
The paper categorifies matroid characteristic polynomials using cohomology.
problem Categorifying matroid characteristic polynomials.
method Using quasi-representations, the paper constructs cohomology groups for matroids.
result The cohomology theory generalizes chromatic and characteristic cohomologies.
Examples of foliations with infinite cohomology dimensions.
problem Understanding foliations with infinite dimensional cohomology.
method Presenting specific foliations and a general condition for their infinite cohomology.
result Foliations have infinite dimensional basic symplectic and complex cohomologies.
Unified theory of orbifolds and cohomology.
problem Formulating a general theory of orbifolds unifying differential and equivariant cohomology.
method Abstract axiomatization in higher topos theory and concrete models for various orbifolds.
result Fully faithful embedding of orbifolds into a cohesive infinity-topos with proper equivariant cohomology.
New cohomology theory for diffeological spaces developed.
problem Cohomology of diffeological spaces.
method Diffeological Čech cohomology theory.
result Established connections between diffeological Čech cohomology and de Rham cohomology.
This study introduces a unified cohomology theory for braided algebras.
problem Classifying infinitesimal deformations of braided algebras.
method Developed a cohomology theory unifying Hochschild and Yang-Baxter cohomology.
result The second cohomology group classifies infinitesimal deformations of braided algebras.
Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.
problem Computing cohomology for foliated manifolds, especially when non-Hausdorff.
method Develops a Künneth formula for specific cases of Hausdorff foliated cohomology and finite-dimensional cohomology.
result Valid Künneth formula for certain foliated cohomology spaces, with counterexamples for others.
The blow-down map is studied in Lie algebroid cohomology.
problem Computing Lie algebroid cohomology of blowups.
method Developed a Gysin sequence for Lie algebroids and used it to compute cohomology.
result Generalized Mazzeo-Melrose theorem to Lie algebroids.
We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilins…
The paper introduces cohomology of coinvariant differential forms and studies its relations with other cohomologies.
problem Understanding the cohomology of coinvariant differential forms and its connections to other cohomologies.
method Defined cohomology of Γ-coinvariant forms and studied its relations with de Rham cohomology and invariant forms cohomology.
result Established relations between cohomology of coinvariant forms, de Rham cohomology, and cohomology of invariant forms.
New cohomological obstruction found for astheno-Kahler metrics.
problem Existence of astheno-Kahler metrics
method New cohomological obstruction
result Found a new cohomological obstruction
Researchers redefine ℓ∞-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.
problem Characterizing groups using ℓ∞-cohomology. method Revisiting Gersten's ℓ∞-cohomology, providing characterizations of amenability and hyperbolicity, and considering algorithmic problems. result Undecidability of some algorithmic problems concerning ℓ∞-cohomology. Inequalities for symplectic cohomology groups are derived.
problem Symplectic cohomology inequalities
method Morse-type inequalities for symplectic Bott-Chern and Aeppli cohomology groups
result Derived inequalities for symplectic cohomology groups
Researchers compute cohomology for a specific type of group action.
problem Computing the equivariant cohomology ring for cohomogeneity-one actions.
method Rational Borel equivariant cohomology computation.
result Rational cohomology ring of a cohomogeneity-one action computed.
Extends Adams' theorem to periodic cohomology.
problem Proving Adams' theorem for periodic cohomology.
method Adapting Adams' approach to periodic cohomology.
result Conjecture proven in a special case.
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
Study shows vast rational cohomology in moduli space of curves with level structures.
problem Understanding the cohomology of moduli spaces with level structures.
method Proved existence of enormous rational cohomology in cohomological dimension.
result Cohomological dimension of moduli space of curves is at least g-2.
Defines log Floer cohomology for symplectic surfaces with a degenerate part.
problem Extending Floer cohomology to degenerate symplectic structures.
method Definition of log Floer cohomology for oriented log symplectic surfaces.
result Log Floer cohomology is invariant under isotopies and isomorphic to log de Rham cohomology for a single Lagrangian.
We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its comput…
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
Cohomology fractals illustrate complex 3-manifold properties.
problem Visualizing complex cohomology classes in hyperbolic 3-manifolds.
method Ray-tracing cohomology fractals and proving their distribution.
result Cohomology fractals converge to a distribution on the sphere at infinity.
Study on twisted Dolbeault cohomology in Kähler foliations.
problem Exploring cohomology in transverse Kähler foliations.
method Analysis of twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem.
result Proved Kodaira-Serre type duality for twisted basic Dolbeault cohomology.
Cohomology defines hyperbolic spaces and their subgraphs.
problem Characterizing hyperbolic spaces and their subgraphs.
method Complete cohomological characterization using ℓ∞-cohomology. result Cohomology vanishing characterizes hyperbolicity and acylindrical hyperbolicity.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
New links show arbitrarily large torsion in Khovanov cohomology.
problem Understanding large torsion in Khovanov cohomology.
method Constructing specific links with desired torsion.
result Found direct summands of arbitrarily large torsion in Khovanov cohomology.
Introduces a new periodic Deligne cohomology with twists by gerbes.
problem Extending Deligne cohomology to higher degrees and periodic forms.
method Combining periodic integral cohomology and Rham complex twists.
result Demonstrates the existence of a twist by gerbes of any odd degree.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
This paper extends group cohomology to bounded cohomology using quasihomomorphisms.
problem Explicitly computing bounded cohomology is hard, especially for groups.
method Interpreting bounded cohomology in terms of group extensions using quasihomomorphisms.
result Makes the interpretation of bounded cohomology in terms of group extensions available.
In this note we study a new cohomology attached to a function along the leaves of complex foliations. We also explain how this cohomology depends on the function and we study a relative cohomology and a Mayer-Vietoris sequence related to this cohomology.
On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both the ordinary smooth Deligne cohomology and the ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant ge…