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.
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…
Paper calculates Torelli group's cohomology second group.
problem Calculating the second rational cohomology group of the Torelli group.
method Building on Hain's and Kupers-Randal-Williams's work, the paper provides an exposition of prerequisite material and the two key results.
result Calculation of the second rational cohomology group of the Torelli group.
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…
The paper studies Fox pairings of Poincaré duality groups using group cohomology.
problem Understanding Fox pairings of Poincaré duality groups.
method Using group cohomology, the paper computes cohomology groups of Fox pairings.
result The paper suggests fundamental and higher Fox pairings.
Paper computes rational cohomology of spin hyperelliptic mapping class groups.
problem Computing rational cohomology of spin hyperelliptic mapping class groups.
method Computes the G-invariant part of the rational cohomology of the pure braid group. result Includes rational cohomology of spin hyperelliptic mapping class groups of genus g. Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
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.
New cohomology theory reveals Q/Z in group homology.
problem Understanding torsion in group homology of diffeomorphism groups.
method Introduced configured group cohomology, yielding explicit R/Z-valued 3-cocycles. result Found a subgroup isomorphic to $\Q/\Z$ in the third group homology of certain diffeomorphism groups.
For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.
The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.
problem Stability properties of bounded cohomology in mapping class groups and symplectic groups.
method Utilizes results from Bestvina and Fujiwara, calculates norms of signature classes, and estimates cohomology norms.
result The bounded cohomology of mapping class groups does not stabilize, while that of symplectic groups does not stabilize via isometries.
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.
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.
We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, wh…
Cohomology of Lie group quotient equals Lie algebra cohomology.
problem Cohomology of Lie group quotients.
method Diffeological de Rham cohomology and Lie algebra cohomology.
result Cohomology of G/H equals Lie algebra cohomology of g/h. Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.
problem Understanding bounded cohomology of groups with prescribed local actions.
method Proving vanishing or infinite bounded cohomology based on the 2-transitivity of F′. result Vanishing or infinite bounded cohomology depending on F′'s 2-transitivity. New computations show various properties of bounded cohomology in finitely presented groups.
problem Understanding bounded cohomology properties in finitely presented groups.
method Computational and theoretical analysis of bounded cohomology.
result Existence of finitely presented non-amenable boundedly acyclic groups and groups with uncountable bounded cohomology.
The paper explores relationships between quandle cohomology, extensions, and automorphisms.
problem Understanding the structure of quandle extensions and automorphisms.
method Establishes a four-term exact sequence and proves relationships involving cohomology, extensions, and automorphisms of quandles.
result Derives new relationships between quandle cohomology, extensions, and automorphisms.
The study generalizes cohomology results for hyperbolic groups.
problem Understanding cohomology of hyperbolic groups and their subgroups.
method Generalizing Gersten's theorem on ℓ∞-cohomology and applying it to hyperbolic groups. result Obtained hyperbolicity criteria for specific groups.
Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic `non-positively curved' groups. On the other hand, there is a well-known interpretation of or…
We give an explicit formula for the cohomology of a right angled Artin group with group ring coefficients in terms of the cohomology of its defining flag complex.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
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. New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
Heaps are para-associative ternary operations bijectively exemplified by groups via the operation (x,y,z)↦xy−1z. They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories…
In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact to…
Study cohomology of homeomorphisms and diffeomorphisms of manifolds.
problem Determine bounded and unbounded cohomology of homeomorphism and diffeomorphism groups.
method Analyzing specific manifolds like the circle, 2-disc, and spheres.
result Identify the bounded cohomology of homeomorphisms and diffeomorphisms groups of certain manifolds.
New method classifies Heintze groups using Lp-cohomology.
problem Quasi-isometry classification of Heintze groups.
method Introducing relative Lp-cohomology and applying it to Heintze groups. result Explicit construction of non-zero relative Lp-cohomology classes. The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the de…
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.
Complex group cohomology surprisingly simple.
problem Computing the cohomology of a complex group's centralizer.
method Explicit computation of rational cohomology.
result Rational cohomology of universal centralizer coincides with that of a point.
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
Study confirms optimal bounds for group cohomology of Lie groups.
problem Optimal bounds for group cohomology of Lie groups.
method Combining complementary vanishings with spectral sequences and quasi-isometry invariance.
result Non-vanishing of group Lp-cohomology for large p and equal degree to rank. In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module A→1. The cohomology of the Lie 2-groups corresponding to the universal crossed modules $G\to \Aut(G)$ and $G\to \Aut^+(G)$ is the abutment of …
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of gro…
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…
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…
Let G be a group that admits a cocompact classifying space for proper actions X. We derive a formula for the Bredon cohomological dimension for proper actions of G in terms of the relative cohomology with compact support of certain pairs of subcomplexes of X. We use this formula to compute the Bredon cohomologi…
Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
Study mapping class group action on cohomology of SL_n character variety.
problem Mapping class group action on cohomology of SL_n character variety.
method Relative endoscopic decomposition and Looijenga's results.
result Reduced problem to cohomology of a finite cover of surface.
Study on cohomology of spin hyperelliptic mapping class groups.
problem Cohomology of spin hyperelliptic mapping class groups.
method Study of G-invariant part of rational cohomology of pure braid groups. result Independence of cohomology dimensions in low degrees and formulas for dimensions.
Novikov conjecture reduced to Lipschitz cohomology of groups.
problem Novikov higher signature conjecture for groups.
method Introducing Lipschitz cohomology classes and reducing the conjecture.
result Reduced Novikov conjecture to Lipschitz cohomology.
New PROP structure defined for automorphism group cohomology.
problem Cohomology of automorphism groups of free groups with specific coefficients.
method Definition of wheeled PROPs and construction of morphisms.
result Construction of morphism φ: E → H with φ(E) = h_1.
Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.
problem Computing the cohomology of mapping class groups with level structures and Prym representations.
method Using twisted cohomology and Prym representations for any positive integer r.
result Cohomology exhibits instability for large genus, but remains stable for r=0 or r=1.
The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We det…
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group G (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, Out(Fn), \dots) embeds via the natural restriction maps into the inverse limit of the secon…