arXiv research

Geometric cohomology model uses co-oriented maps to define a product structure.

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

The abstract discusses p-harmonic forms and their geometric properties, proving new theorems about Lp-cohomology.

Study on geometrically formal metrics on complex manifolds.

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…

Notes describe geometric interpretations of cohomology in trisected 4-manifolds.

New cohomology theory reveals $\mathbb{Q}/\mathbb{Z}$ in group homology.

We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.

Just as $\Cstar$ principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral cohomology through their Dixmier-Douady class. I consider an alternative, related, geometric realisation of three dimen…

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…

Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.

The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.

Introduces quantum cohomology and helices in geometric methods.

We study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.

We describe the geometrical ladder of equations for Abelian bundles and gerbes, as well as higher generalisations, in terms of the cohomology of an operator that combines de Rham and Cech cohomology.

This paper extends geometric structure theory to infinite type structures.

This is a short survey of Riemannian geometric applications of Lp-cohomology of thick spaces, p not equal to 2.

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.

We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the $L_{q,p}$-cohomology of these manifolds. Some applications to vanishing and non vanishing results in $L_{q,p}$-cohomology are given.

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and …

Unified theory of orbifolds and cohomology.

Link homology compared with geometric link invariants using Bott-Samelson varieties.

New geometric model for knot homology using monodromic Hecke category.

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of $H$-thick knots which are as far of being $H…

We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.

Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends…

Introduces positivity for $(1,1)$ classes in foliated manifolds.

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…

Paper calculates stable cohomology of universal degree d hypersurfaces.

Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.

We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study h…

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…

The paper introduces a method to decorrelate circular coordinates using lattice reduction.

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…

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been expl…

For product manifolds, cohomologically calibrated affine connections are geometrically irreducible.

Cohomology fractals are visual representations of cohomology classes on hyperbolic 3-manifolds.

Develops Hodge theory for boundary-value problems on general geometric structures.

Novel Morse theory for mapping cone cohomology.

The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which accommodates the KdV equation and other well known integrable systems. In the case of qua…

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{…

We describe the unitary globalization of cohomologically induced modules $A_{\fq}(λ)$. The purpose of the paper is to give a geometric realization of the unitarizable modules. Our results do not constitute a proof of unitarity.

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.

Researchers redefine $\ell^\infty$-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.

Introduces Lie-Yamaguti algebra bundles and their cohomology.

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rat…