arXiv research

Uniform criterion for vanishing products in bounded cohomology.

Extends cohomology theory for infinite volume transformation groups.

New proofs and refined theorems on bounded cohomology.

Injective construction proves bounded cohomology dimensions.

Study cohomology of homeomorphisms and diffeomorphisms of manifolds.

New computations show various properties of bounded cohomology in finitely presented groups.

Constructs Serre spectral sequence for bounded cohomology.

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomolog…

New displacement technique vanishes bounded cohomology in all degrees.

Trivial Massey product in specific cohomology groups.

Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the d…

The paper studies knot quandles and their cohomology, proving infinite dimensionality results.

Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.

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…

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…

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded …

Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.

The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.

We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is d…

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, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the secon…

Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reduc…

New techniques prove bounded acyclicity results for semi-simplicial sets.

Study of quasimorphisms and bounded cohomology in braided Thompson groups.

Two main theorems are proved in this paper. Theorem 1: There is a constant C(n, D) depending only on n and D such that for a closed Riemannian n-manifold satisfying Ric > -(n-1) and Diam < D, the ith bounded Betti number is bounded by C(n, D). Here the ith bounded Betti number is defined as the dimension of the image o…

Relative cup-length defined for non-Morse functions on manifolds.

Study cohomology of surfaces with punctures and boundaries, proving bounds on rational cohomology.

New bounded cohomology classes found for exact forms on curved manifolds.

Study shows non-abelian free groups' 4th cohomology is non-zero.

Develops Lefschetz theory for noncompact manifolds.

The paper introduces new functors for cohomology groups of manifolds.

Generalizes integration map to coinvariants of bounded functions.

We introduce a "qualitative property" for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the $\partial\overline\partial$-Lemma. This follows from a quantitativ…

The paper proves bounded cohomology properties of Euclidean space groups.

The paper bounds Betti numbers of complex-hyperbolic manifolds.

The paper studies cohomology of groups with contracting elements.

Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,μ)$, $Diff_0(M,vol)$ and $Symp_0(M,ω)$ (in case $M$ is symplectic). As an application we show that, under certain conditio…

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…

The paper studies cohomology of groups acting on 1-manifolds and applies results to spectrum problems.

Boundary of fiber convex domains is a cohomological sphere.

Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.

We show that the isomorphism induced by the inclusion of pairs $(X,\emptyset)\subset (X,Y)$ between the relative bounded cohomology of $(X,Y)$ and the bounded cohomology of $X$ is isometric in degree at least 2 if the fundamental group of each connected component of $Y$ is amenable. As an application we provide a self-…

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.

Let $\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let ${E_2^{>0,\bullet}}=\oplus\{E_2^{p,q}\colon p>0,q\geq 0\}$ be the subalgebra of cohomology classes with positive transverse degree in the $E_2$ term of the spectral sequence of the foliation. We prove that the saturated transverse Lusternik-S…

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…

Study Witten deformation on noncompact manifolds with bounded geometry.

The paper proves vanishing bounded cohomology for various groups.