arXiv research

Examples of foliations with infinite cohomology dimensions.

Extends cohomology theory for infinite volume transformation groups.

A manifold's co-invariant cohomology can be infinite-dimensional.

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

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

Study anti-invariant cohomology on almost complex manifolds, showing infinite and finite dimensions.

Study mapping class groups of infinite type surfaces, classify loxodromic elements, and prove infinite-dimensional cohomology.

We construct several infinite families of nonnegatively curved manifolds of low cohomogeneity and small dimension which can be distinguished by their cohomology rings. In particular, we exhibit an infinite family of eight-dimensional cohomogeneity one manifolds of nonnegative curvature with pairwise non-isomorphic comp…

Study calculates integral cohomology of non-orientable infinite type surfaces.

The paper studies how Kleinian groups' bounded cohomology distinguishes hyperbolic 3-manifolds' ends.

This paper generalizes L2 cohomology theory for complex manifolds.

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

New method constructs bounded cohomology classes for specific transformation groups.

Study Euler class of surface bundles with nontrivial results.

The paper studies cohomology of groups with contracting elements.

New insights into mapping class groups' cohomology.

This paper extends geometric structure theory to infinite type structures.

Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.

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 of quasimorphisms and bounded cohomology in braided Thompson groups.

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 extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW complexes. This pa…

In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.

Study of pure mapping class groups on infinite graphs.

We find a splitting in a special cohomology theory for complex manifolds.

Consider a complete orientable manifold with countably many components of bounded dimension. Suppose that its rational homology is infinitely generated in some degree. Then there is no choice of weight function for which the natural map from weighted L^2 cohomology to de Rham cohomology is surjective in that degree.

Injective construction proves bounded cohomology dimensions.

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…

Let $M$ be a compact abstract $CR$ manifold of arbitrary $CR$ codimension. Under certain conditions on the Levi form we prove the infinite dimensionality of some global cohomology groups of $M$.

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 cohomology of mapping class groups for big genus surfaces.

Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.

The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.

New quandles classify surfaces, with infinite cohomology.

We define algebraic structures on graph cohomology and prove that they correspond to algebraic structures on the cohomology of the spaces of imbeddings of S^1 or R into R^n. As a corollary, we deduce the existence of an infinite number of nontrivial cohomology classes in Imb(S^1,R^n) when n is even and greater than 3. …

In this article, we present new symplectic 4-manifolds with same integral cohomology as $S^{2}\times S^{2}$. The generalization of this construction is given as well, an infinite family of symplectic 4-manifolds cohomology equivalent to $#_{(2g-1)}{(S^{2}\times S^{2})}$ for any $g\geq 2$. We also compute the Seiberg-Wi…

Infinite families of quantum modular invariants for 3-manifolds are discovered.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

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…

Study cohomology of GL₂n(Z) and graph complexes using Pfaffian forms.

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined an…

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…

For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G it decomposes into a sum of irreducible components, each of which appears in it …

Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeom…

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…

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 $n \ge 3$.

We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for the configuration spaces of three collinear and non-collinear points.