Proves subgroups of relatively hyperbolic groups with Bredon cohomological dim 2 are also relatively hyperbolic.
problem Proving subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2 are also relatively hyperbolic.
method Algebraic approach to relative homological Dehn functions and characterization of relative hyperbolicity over Bredon modules.
result Proves the analogous result for relatively hyperbolic groups of Bredon cohomological dimension 2.
In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological…
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…
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.
Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n≥3.
Bredon's trick helps extend local properties to global topological spaces.
problem Extending local properties to global topological spaces.
method Bredon's trick for local properties to global spaces.
result Bredon's trick allows for natural alternative demonstrations of classic results.
Computes virtually cyclic dimension for 3-manifold groups.
problem Computing the virtually cyclic geometric dimension of 3-manifold groups.
method Prime and JSJ decompositions of M, push-out type constructions, Bredon cohomology computations.
result Explicit computation of virtually cyclic geometric dimension for 3-manifold groups.
Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
Let T be a torus. We present an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space. This sequence, which goes back to Atiyah and Bredon, generalizes the so-called Chang-Skjelbred lemma. As coefficients, we allow prime fields and subrings of the rationals, i…
We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness o…
Compute Bredon homology for a specific type of Artin groups.
problem Calculate Bredon homology for Artin groups of dihedral type.
method Compute Bredon homology groups of the classifying space for virtually cyclic subgroups with K-theory coefficients.
result Computed Bredon homology groups for Artin groups of dihedral type.
Researchers compute K-theory for cohomogeneity-one actions.
problem Computing equivariant K-theory for cohomogeneity-one actions.
method Equivariant homotopy theory, representation theory, Lie theory.
result Derived generators and relations for K-theory ring.
We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of …
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
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.
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…
Study shows vast rational cohomology in moduli space of curves with level structures.
problem Understanding the cohomology of moduli spaces with level structures.
method Proved existence of enormous rational cohomology in cohomological dimension.
result Cohomological dimension of moduli space of curves is at least g-2.
We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem
The paper estimates the dimension of cohomology for semipositive line bundles on various manifolds.
problem Estimating the dimension of cohomology for semipositive line bundles over different types of manifolds.
method Asymptotic estimates for harmonic (0,q)-forms with high tensor powers of semipositive line bundles. result Asymptotic estimates for the dimension of cohomology of semipositive line bundles on various manifolds.
Study on cohomological dimension of surface terms, answering Farb's question.
problem Determining the cohomological dimension of specific surface terms.
method Analyzing the Johnson filtration of closed, orientable surfaces of genus g≥2. result The kth term of the Johnson filtration has cohomological dimension 2g−3 for all k≥3 and g≥2. Lattices in Lie groups have a geometric dimension equal to their cohomological dimension.
problem Understanding the geometric structure of lattices in Lie groups.
method Proved homotopy equivalence to a CW complex.
result Symmetric space dimension equals virtual cohomological dimension of the lattice.
The paper studies embeddings and cohomological dimensions of braid and mapping class groups of surfaces.
problem Understanding embeddings and cohomological dimensions of braid and mapping class groups of surfaces.
method Utilizing the relationship between braid and mapping class groups of a surface and its orientable double covering, the paper generalizes results and computes cohomological dimensions.
result Upper bounds for the virtual cohomological dimension of MCG(N ; k) are given if the genus of N is greater than or equal to 2.
New triangulations of octonionic projective plane found with restricted symmetry groups.
problem Finding symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane.
method Using Smith and Bredon's results on transformation groups to restrict possible symmetry groups.
result List of 26 subgroups of S27 containing all possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic project plane.
Generalizes Alexandroff's Vn-continua to cohomological dimensions.
problem Extending Alexandroff's concept of Vn-continua to cohomological dimensions. method Proves that strongly locally homogeneous generalized continua with cohomological dimension n are generalized Vn-spaces. result Every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. Generalizes theorem for topological G-manifolds with linear Lie groups G.
problem Understanding topological G-manifolds with linear Lie groups G. method Using countable CW complexes and Palais-proper actions.
result Topological G-manifolds have G-homotopy type of countable G-CW complexes. Thurston's spine dimension exceeds virtual cohomological dimension.
problem Understanding the dimension of Thurston's spine in moduli space.
method Analyzing closed hyperbolic surfaces and their systoles.
result The dimension of the set of surfaces is at least \(5-\varepsilon\) for large genus.
New contractible complex shows virtual cohomological dimension of RAAGs.
problem Determining virtual cohomological dimension of RAAGs.
method Equivariant deformation retraction of spine to contractible cube complex.
result Dimension of new complex realises virtual cohomological dimension of RAAGs.
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
Klein bottle embeds into specific lens spaces.
problem Embeddability of Klein bottle in lens spaces.
method Direct proof and explicit realizations.
result Klein bottle embeds into L(4n,2n±1) only. Study cohomological properties of complex manifolds, extending a result to higher dimensions.
problem Upper bound for Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces.
method Suitable metric conditions, extending a result by A. Teleman to higher dimensions.
result Upper bound for Bott-Chern cohomology in terms of Betti numbers for compact complex manifolds.
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SLn(Z) at virtual cohomological dimension. method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
Sharp bound on cohomology of quasiregularly elliptic manifolds.
problem Bounding cohomological dimension of manifolds with quasiregular mappings.
method Using combinatorial and geometric properties of cohomology, proving a sharp upper bound.
result A sharp upper bound on the cohomological dimension of manifolds, answering Gromov's question.
Study on Lp cohomology and Hodge decomposition for ALE manifolds.
problem Understanding Lp cohomology dimensions and harmonic forms in ALE manifolds. method Relating dimensions of Lp cohomology spaces to decaying harmonic forms, proving independence and jumps in dimensions, and providing Hodge decompositions. result Dimension of Lp reduced cohomology spaces in degree k is independent of p for k not equal to 1 or n-1, and jumps by a factor N-1 for k equal to 1 or n-1. 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.
We study Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms whose square is zero. In particular, we prove an inequality à la Frölicher relating the dimensions of the Bott-Chern and Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove that the equa…
Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described…
Study shows flat torus is identified by first cohomology group's dimension.
problem Identifying flat torus among RCD∗(0,N) spaces. method Analysis of the first cohomology group.
result Proves flat torus when first cohomology group dimension equals N.
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.
Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.
Study calculates Poisson cohomology of broken Lefschetz fibrations.
problem Computing Poisson cohomology of broken Lefschetz fibrations.
method Calculates cohomology at fold and Lefschetz singularities, adapting techniques for Sklyanin algebra.
result Compact formulas for Poisson coboundary operator in 4 dimensions.
Estimates cohomology dimension via Kato constant for compact manifolds with integral Ricci bounds.
problem Estimating the dimension of the first cohomology group on compact manifolds.
method Using Ricci curvature integral assumptions and Kato constant of the negative part of Ricci curvature.
result Quantitative statements about the cohomology group are provided, contrary to previous results.
Proves existence and regularity of minimizers for surfaces in arbitrary dimensions.
problem Existence and regularity of minimizers for surfaces in arbitrary dimensions.
method Generalizes and extends methods of Reifenberg, Besicovitch, and Adams, proving existence of minimizing sequences and developing cohomological spanning conditions.
result Existence and regularity of minimizers for surfaces in arbitrary dimensions, satisfying cohomological boundary conditions.
Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.
problem Computing Čech cohomology groups of Morse boundaries.
method Analyzing cusped hyperbolic n-manifolds and relatively hyperbolic groups.
result Reduced Čech cohomology vanishes in dimensions ≤ n-3 and does not vanish in dimension n-2.
Study cohomology of surfaces with punctures and boundaries, proving bounds on rational cohomology.
problem Understanding cohomology of surfaces with punctures and boundaries.
method Two proofs showing congruence subgroups have enormous rational cohomology.
result Bounds on cohomology are super-exponential in number of punctures and boundary components.
Study quaternionic Bott-Chern cohomology and HKT metrics existence.
problem Existence of HKT metrics on hypercomplex manifolds.
method Adapt results from complex geometry to quaternionic setting and prove a criterion.
result Criterion for existence of HKT metrics on compact hypercomplex manifolds of real dimension 8.
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.