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.
Direct product cohomological dimension equals twice the group's dimension.
problem Cohomological dimension of direct product groups
method Proved geometrically finite groups cohomological dimension
result Direct product cohomological dimension equals 2 times the group's dimension
Computes cohomology of smooth cubic surfaces using simplicial resolution.
problem Computing the cohomology of smooth cubic surfaces.
method Vassiliev's simplicial resolution method.
result Cohomology isomorphic to that of P2 modulo embedding. We prove properties of the Schweitzer complex and its cohomologies.
problem Properties of the Schweitzer complex and its cohomologies.
method Elliptic complex and cohomological functors.
result Finite dimensionality and equivalence to Hirzebruch-Riemann-Roch relations.
Study cohomology groups of specific Alexander f-quandles over finite fields.
problem Determining cohomology groups for a specific class of Alexander f-quandles.
method Analyzing the cohomology groups of Alexander f-quandles of the form Fq[T,S]/(T−ω,S−β). result Computed the second, third, and fourth cohomology groups.
In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere Sn using known results about the cohomology of the symmetric product of spheres.
The paper constructs symplectic forms on frame bundles and proves finite cohomologies.
problem Proving finiteness theorems for basic symplectic cohomologies.
method Construction of invariant 2-forms on the real symplectic group, symplectic form on quotient by a maximal torus, and lifting symplectic structure to frame bundles.
result Valid proof of finiteness theorems for basic symplectic cohomologies.
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
Study finite groups acting on 3D shapes with rational homology.
problem Identify finite groups that can act freely on rational homology 3-spheres.
method Cohomology of groups to analyze finite group actions.
result Characterize finite groups that can act freely and trivially on rational homology 3-spheres.
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. Study shows intersections of stable subgroups have finite properties.
problem Understanding intersections of stable subgroups in finitely generated groups.
method Analyzing intersections of conjugates and extending quasimorphisms.
result Finite families of quasimorphisms can be simultaneously extended to the ambient group.
New method constructs bounded cohomology classes for specific transformation groups.
problem Constructing classes in bounded cohomology of transformation groups.
method New method of constructing classes in bounded cohomology of transformation groups.
result 3rd bounded cohomology of certain groups is infinite dimensional.
We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space X of boun…
Integrable Pfaffian systems invariant under Lie group actions have their cohomology equal to Lie algebra cohomology.
problem Detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.
method Using Lie algebra cohomology to compute and detect obstructions in the variational cohomology of invariant Pfaffian systems.
result The vertical variational cohomology of an invariant Pfaffian system is equal to the Lie algebra cohomology of the Lie algebra g. 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.
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.
New spectral sequence helps determine when cohomology of configuration spaces is finite.
problem Determining when cohomology of configuration spaces is finitely generated.
method Derived a spectral sequence from the homological algebra of the partition lattice.
result Criterion for finitely generated mFI module Hi(mConfn(X)). We compute the compactly supported cohomology of the standard realization of any locally finite building.
Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
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.
Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
We prove that the basic intersection cohomology IHpˉ∗(M/F), where F is the singular foliation determined by an isometric action of a Lie group G on the compact manifold M, is finite dimensional.
Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.
We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.
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…
We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. W…
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.
problem Identifying hyperbolic 3-manifolds using their finite quotient groups.
method Proves profinite isomorphism of fundamental groups uniquely determines the manifold's isomorphism type.
result Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups.
The paper explores cohomologies on almost complex manifolds and their applications.
problem Distinguishing and understanding cohomologies on non-integrable almost complex structures.
method Defined and studied cohomologies HN∙(M) and HJ∙(M) using Nijenhuis-Lie derivations. result The J-cohomology encodes whether an almost complex structure satisfies the dLJ-lemma. We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagr…
We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
The Burde--de Rham theorem is extended to finitely presented pro-p groups with specific conditions.
problem Extending the Burde--de Rham theorem to pro-p groups with certain constraints. method Assumption of total degrees of relators being 0, concrete examples, and cohomological interpretations.
result The theorem is extended to finitely presented pro-p groups under specified conditions. Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
Study finite group actions on 4-manifolds, finding rank bounds.
problem Understanding finite group actions on 4-manifolds.
method Investigate Borel spectral sequence for G-equivariant cohomology.
result Establish new bounds on the rank of G for homologically trivial actions.
We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-com…
Study cohomology jump loci in 3-manifolds, focusing on Alexander polynomial.
problem Understanding cohomology jump loci in 3-manifolds and links.
method Analysis of characteristic and resonance varieties, using Alexander polynomial.
result Derive consequences on formality and finite-dimensional models.
In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen's geometric description of cobordism groups for finite dimensional smooth manifolds \cite{Q}. Quillen stresses the fact that this construction allows the definition of Gysin ma…
Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies…
We obtain the cohomology of the variational bicomplex on the infinite order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. This provides a solution of the global inverse problem of the calculus of variations of finite order on fiber bundles.
The paper studies cohomology of groups acting on 1-manifolds and applies results to spectrum problems.
problem Understanding cohomology of groups acting on 1-manifolds and its applications to spectrum problems.
method Proves a criterion for vanishing second bounded cohomology and applies it to various groups and spectrum problems.
result Provides new computations of second bounded cohomology and solves several spectrum problems.
Method implements symmetries in TQFT for finite groups.
problem Implementing symmetries in TQFT for finite groups.
method Cohomology classes and bordisms with a (d−q−2)-skeleton. result Generalizes existing methods to arbitrary dimensions and q.
Let G be a simply connected solvable Lie group with a lattice Γ and N the nilradical of G. For a complex valued representation ρ:G→GL(Vρ) such that the restriction ρ∣N is unipotent, as an advanced variation of cohomology computation of solvmanifolds by using Lie algebra cohomology, we construct a…
For a simply connected (non-nilpotent) solvable Lie group G with a lattice Γ the de Rham and Dolbeault cohomologies of the solvmanifold G/Γ are not in general isomorphic to the cohomologies of the Lie algebra g of G. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfr…
Study stabilizes arithmetic statistics of rational maps over finite fields.
problem Stability of arithmetic statistics of rational maps over finite fields.
method Representation stability and arithmetic statistics of spaces of 0-cycles.
result Arithmetic quantities associated to rational maps over finite fields stabilize as degree increases.
Extends Lannes-Quillen theorem to all profinite groups.
problem Conjugacy separability of p-torsion elements and finite p-subgroups. method Developed a theory of products for families of discrete torsion modules.
result Proves a full version of the Lannes-Quillen theorem for all profinite groups.
Generalizes group cohomology and holonomy Lie algebra computations.
problem Computing cup-products and holonomy Lie algebras for arbitrary groups.
method Explicit formula for cup-products, Magnus expansion method for holonomy Lie algebras.
result Explicit formula for cup-products in cohomology of 2-complexes.