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.
New result on embedding cohomology of hyperbolic groups.
problem Embedding cohomology of hyperbolic groups into their virtually free subgroups.
method Probabilistic argument and inverse limit of cohomologies.
result Second bounded cohomology of acylindrically hyperbolic groups embeds into cohomologies of virtually free subgroups.
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.
The paper studies knot quandles and their cohomology, proving infinite dimensionality results.
problem Understanding the cohomology of knot quandles and its implications in knot theory.
method Analyzing quandles and their inner automorphism groups, proving conditions for infinite dimensionality.
result The second bounded cohomology of knot quandles is infinite dimensional, detecting the unknot.
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 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.
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.
Trivial Massey product in specific cohomology groups.
problem Massey products in bounded cohomology.
method Analyzing specific groups and forms.
result Massey triple product is trivial under given conditions.
Introduces new cohomology theories for Lie 2-algebras and groups.
problem Classical cohomology theories do not extend to Lie 2-algebras and groups.
method Develops new cohomology theories and uses them to prove integrability.
result New cohomology theories classify extensions and prove integrability of Lie 2-algebras.
New computations for third and fourth cohomology of Lie group spaces.
problem Computing cohomology groups of homogeneous spaces of Lie groups.
method Explicit descriptions of third and fourth real de Rham cohomologies in terms of Lie-theoretic data.
result For a large class of homogeneous spaces, the difference between third and fourth Betti numbers equals the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.
The paper studies cohomology of groups with contracting elements.
problem Understanding the cohomology of groups with specific elements.
method Proving infinite-dimensional relative bounded cohomology for groups with contracting elements.
result The cohomology is infinite-dimensional for groups with contracting elements.
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. The Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of tw…
Continuous cohomology theory for topological quandles introduced and compared.
problem Developing a continuous cohomology theory for topological quandles.
method Introduced continuous cohomology theory, compared to algebraic theories, studied extensions with continuous 2-cocycles, computed cohomology groups of inverse limits.
result Showed differences in second cohomology groups for specific topological quandles.
The paper proves vanishing cohomology groups for free boundary hypersurfaces.
problem Proving vanishing cohomology groups for free boundary hypersurfaces.
method Using a universal constant and traceless second fundamental form condition.
result The pth cohomology group of a compact free boundary submanifold vanishes. Study mapping class groups of infinite type surfaces, classify loxodromic elements, and prove infinite-dimensional cohomology.
problem Classifying elements in mapping class groups of infinite type surfaces.
method Classify loxodromic elements with WWPD action on loop graphs.
result Prove infinite-dimensional second bounded cohomology for certain subgroups.
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
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.
The paper computes the de Rham cohomology of real flag manifolds.
problem Computing the second de Rham cohomology group of real flag manifolds.
method Using the Weil construction and computations of the second homology group.
result The second de Rham cohomology group is zero in general, with some exceptions.
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
This note proves equivariant de Rham cohomology for quotient spaces.
problem Computing de Rham cohomology of quotient spaces under group actions.
method Equivariant identification of de Rham complexes using foliation theory.
result Canonical isomorphism of de Rham complexes for quotient spaces.
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.
The paper calculates cohomology of complex manifolds with cyclic group actions.
problem Calculating the integral cohomology of complex manifolds with cyclic group actions.
method Investigates toric blow-ups and uses spectral sequences of equivariant cohomology.
result Provides necessary and sufficient conditions for spectral sequence degeneration.
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.
We compute the cohomology of a Fuchsian group of the second kind with coefficients in the hyperfunction vectors of the principal series representations of SL(2,R) supported on the limit set.
We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry.
Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uni…
Consider a manifold endowed with the action of a Lie group. We study the relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and we show that the cohomology of the Cartan complex lies on the 0-th row of the second page of a …
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.
I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class gr…
We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically non-vanishing in D>1. Hence, in contrast with the D=1 case, the deformation theory in the multivariable case is non-trivial.
Computes cohomology classes of strata of meromorphic differentials.
problem Computing cohomology classes of differentials with prescribed poles.
method Defined a space of stable meromorphic differentials, stratified by zero multiplicities, computed Poincaré-dual cohomology classes, proved tautological, and provided an algorithm.
result All cohomology classes are tautological and can be computed.
We describe the second integral cohomology group of a surface bundle as the group of Chern classes of fiberwise holomorphic complex line bundles and use this to obtain information on this group.
We classify compact oriented 5-manifolds with free fundamental group and π2 a torsion free abelian group in terms of the second homotopy group considered as π1-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply t…
We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb-Kaimanovich-Masur rigidity theorem that states that MCG(S) does not contain a higher rank lattic…
Study Torelli subgroups of handlebody groups and their cohomology.
problem Understanding the cohomology of handlebody Torelli groups.
method Introduce Torelli subgroups, use Johnson homomorphisms, and symplectic representations.
result Describe cup products in the first rational cohomology groups of handlebody Torelli groups.
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.
Graph cohomology solves symplectic problems in surface mapping groups.
problem Compute the symplectic decomposition of Torelli group and its cohomology.
method Graph cohomology and ideas from graph cohomology.
result Effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group.
Smoothness conditions on moduli spaces and character varieties are examined.
problem Conditions for smooth points on moduli spaces of flat connections and character varieties.
method Gauge theoretic and algebraic methods, including slice theorem and Fox calculus.
result Smoothness conditions differ between moduli spaces of flat connections and character varieties.
The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP^1), with coefficients in the space of bilinear differential operators that act on tensor densities, D_{λ, ν;μ}, vanishing on the Lie algebra sl(2,R)…
We prove the Carlson-Toledo conjecture (for all Kahler groups which do not have property T of Kazhdan).We deduce a conjecture of Goldman-Donaldson for all 3-manifold groups which are rich in the sense of [Re2].
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the "current" module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.
This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined wi…
We briefly indicate some implications of [1] for the second Lie algebra cohomology of equivariant map algebras and (twisted multi) loop algebras.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Solves Nielsen realization problem for hyper-Kähler manifolds.
problem Realization problem for hyper-Kähler manifolds.
method Uses same invariant as for K3 surfaces and determines representation of mapping class group.
result Representation of mapping class group admits a section on its image for some deformation types.
This paper introduces persistent equivariant cohomology and applies it to circle actions.
problem Understanding the cohomology of filtered spaces with group actions.
method Persistent Borel equivariant cohomology, Serre spectral sequence, Gysin homomorphism.
result Explicit description and cohomology computation for circle actions.