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.
In this short note we define a new cohomology for a Lie algebroid A, that we call the \emph{twisted cohomology} of A by an odd cocycle θ in the Lie algebroid cohomology of A. We proof that this cohomology only depends on the Lie algebroid cohomology class [θ] of the odd cocycle $…
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.
We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…
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.
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.
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. 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…
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.
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.
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.
Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been expl…
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.
New cohomological obstruction found for astheno-Kahler metrics.
problem Existence of astheno-Kahler metrics
method New cohomological obstruction
result Found a new cohomological obstruction
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…
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 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. We relate Lq,p-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of ℓq,p-cohomology, packing cohomology. This implies quasi-isometry invariance of Lq,p-cohomology together with its multiplicative structure. The result partially extends to the Rumin Lq,p-cohomolog…
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.
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.
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…
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.
In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.
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.
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…
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.
Study on properties of Oeljeklaus-Toma manifolds, including cohomology and metrics.
problem Characterizing and understanding the metric and cohomological properties of Oeljeklaus-Toma manifolds.
method Analysis of double complex of differential forms, Bott-Chern cohomology, and explicit formulas for Dolbeault cohomology.
result Proved that Oeljeklaus-Toma manifolds do not admit certain types of metrics and provided explicit formulas for their Dolbeault cohomology.
Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are c…
New proofs and refined theorems on bounded cohomology.
problem Properties of bounded cohomology and comparison map.
method Homotopy-theoretic properties and generalizations.
result New proofs and refined versions of vanishing and covering theorems.
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ=0, and the algebraic o…
We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certian results on the cohomology of solvmanifolds and infra-solvmanifold…
Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.
problem Dolbeault and Bott-Chern cohomology of Oeljeklaus-Toma manifolds.
method Explicit harmonic representatives and geometric analysis.
result Showed geometric Dolbeault formality and studied Angella-Tomassini inequality.
New cohomology groups generalize Euler number for Lie superalgebras.
problem Generalizing cohomology groups for Lie superalgebras.
method Abstracted Poisson cohomology groups to Poisson-like cohomology groups for general Lie superalgebras.
result De Rham cohomology groups match Poisson-like cohomology groups for differential forms.
Algorithm calculates stable multiplicities in cohomology of configuration spaces.
problem Computing stable multiplicities of irreducible representations in cohomology.
method Developed an algorithm to compute stable multiplicities of families of irreducible representations.
result Computed stable multiplicities for all Young diagrams with 23 boxes up to degree 50.
We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
Cohomology of 'book' Lie algebra Poisson structure computed.
problem Computing the cohomology of a specific Lie algebra structure.
method Direct computation of cohomology for the given Lie algebra structure.
result Explicit formula for Poisson cohomology of the 'book' Lie algebra.
By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.