The paper compares two notions of invariance in Hochschild cohomology spaces.
problem Comparing two notions of invariance in Hochschild cohomology spaces.
method Analyzing the induced action on Hochschild cohomology of smooth functions and comparing two spaces of invariants.
result For proper group actions, both spaces of invariants are isomorphic.
Left-invariant forms solve Dolbeault cohomology for complex nilmanifolds.
problem Computing Dolbeault cohomology for complex nilmanifolds.
method Inducing isomorphisms in cohomology using left-invariant forms.
result Invariant forms compute Dolbeault cohomology in every bidegree.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
Study anti-invariant cohomology on almost complex manifolds, showing infinite and finite dimensions.
problem Understanding the cohomology of anti-invariant forms on almost complex manifolds.
method Construction of specific almost complex structures and analysis of cohomology groups.
result Found manifolds with anti-invariant cohomology of infinite, 0-1, and 2 dimensions.
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
Paper connects cohomologies on almost complex manifolds.
problem Relating J-invariant cohomology to Dolbeault cohomology. method Relating cohomologies through necessary and sufficient conditions.
result Conditions for isomorphism between cohomologies found.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
A manifold's co-invariant cohomology can be infinite-dimensional.
problem Infinite-dimensional co-invariant cohomology of manifolds.
method Defined Γ-co-invariant cohomology and cohomology of divergence forms. result Presented a manifold with infinite-dimensional co-invariant cohomology.
Modified Θ-invariant for non-vanishing cohomology groups.
problem Defining Θ-invariant for non-vanishing cohomology groups. method A slightly modified version of the Θ-invariant. result Can define Θ-invariant even when cohomology group is not vanishing. Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.
New cohomological invariants defined for foliations.
problem Understanding cohomology of foliations under different metrics.
method Defining and analyzing basic and antibasic cohomologies, using Laplace operator.
result Antibasic cohomology groups are invariant under diffeomorphisms and metric changes.
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…
Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer ho…
In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold (M,ξ,η,Φ,g), and conclude the pureness and fullness of Φ-invariant and Φ-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is a…
We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…
Homotopy invariance proven for twisted Lie algebroid cohomologies.
problem Homotopy invariance of twisted Lie algebroid cohomologies.
method Lie algebroid homotopy-invariance proof with examples.
result Comprehensive systematic way to compute twisted Lie algebroid cohomologies.
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.
We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.
The paper studies cohomologies of hypercomplex manifolds and their dimensions.
problem Understanding cohomologies and dimensions of invariant and anti-invariant subgroups.
method Proving a compact hypercomplex manifold is C∞-pure-and-full under certain conditions and studying dimensions of subgroups. result Characterization of hyperkähler with torsion metrics in terms of the dimension of the Jˉ-invariant subgroup. New method simplifies cohomology computation for specific Lie group structures.
problem Computing cohomology for hypocomplex structures on compact Lie groups.
method Dual of Fréchet-Schwartz spaces theory applied to hypocomplex structures.
result Top-degree cohomology can be computed using only left-invariant forms.
We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-Kähler structure (J, ω, g). In particular, almost-Kähler manifolds satisfying a Lefschetz type property, and solvmanifolds endowed with left-invariant almost-complex structures …
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.
Let S be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of S is equal to the Lie …
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.
Deep analysis of KV-Poisson structures and new invariant computed.
problem Understanding KV-Poisson structures and their properties.
method Classification, properties analysis, and computation of Cohomological groups.
result KV-Poisson structures give rise to a new Cohomological invariant.
Study on Kodaira fibrations with nontrivial cohomology, proving properties of their structure.
problem Characterizing Kodaira fibrations with specific cohomology properties.
method Analyzing invariant rational cohomology and properties of holomorphic sections.
result Kodaira fibrations with invariant cohomology admit specific coverings and monodromies.
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
Study on deformations of Lie groupoid morphisms and their properties.
problem Understanding the deformation theory of Lie groupoid morphisms.
method Established deformation theory, cohomology, and properties of morphisms.
result Invariance and stability properties of morphisms, Morita invariance of cohomology, and simultaneous deformations.
The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. Characterizes Spinor groups using cohomology operations.
problem Understanding the cohomology and invariants of Spinor groups.
method Constructing integral cohomology rings and characteristic classes.
result Determined the ring of integral Weyl invariants of Spin(n).
We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…
New spectral sequence for K-manifolds, computing cohomology and harmonic forms.
problem Computing cohomology and harmonic forms of K-manifolds. method Introducing a new spectral sequence and using it to generalize theorems from K-contact geometry. result Computed cohomology ring and harmonic forms of S-manifolds. Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.
problem Calculating characteristic class relations in Dolbeault cohomology.
method Representation theory of structure group, without metric or connection.
result First results on Chern-Simons invariants of Cartan geometries.
Study on Riemannian foliations, decomposing cohomology based on J structure.
problem Decomposition of basic cohomology for specific Riemannian foliations.
method Analysis of codimension four taut Riemannian foliations using J structure.
result Obtained estimates for the dimension of basic J-anti-invariant subgroup.
We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle …
Lattice cohomology, defined by Némethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two inva…
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface Σ and the Donaldson invariants of the algebraic surface Σ×P1. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. …
We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
Researchers prove no unexpected relations between complex manifold numbers.
problem Proving no unexpected universal linear relations between Hodge, Betti, and Chern numbers of compact complex manifolds.
method Developed a framework to tackle more general questions involving all cohomological invariants, solved specific construction problems.
result Obtained full answers to general questions about universal relations and bimeromorphic invariants in low dimensions.
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.