arXiv research

Classifies and computes cohomologies of complex structures on Lie groups.

Unified framework for Morita invariant cohomology of Lie groupoids.

Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perfor…

New cohomology theory shows compact Lie group actions are Morita invariant.

New deformations of lattice cohomology help calculate knot invariants.

Paper connects cohomologies on almost complex manifolds.

Quantum cocycle invariants derived from Yang-Baxter cohomology.

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

Study new invariants in complex geometry using Bott-Chern hypercohomology.

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…

Study cohomological and metric properties of non-Kähler complex manifolds.

We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer ho…

In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold $(M,ξ,η,Φ,g)$, and conclude the pureness and fullness of $Φ$-invariant and $Φ$-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is a…

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…

Homotopy invariance proven for twisted Lie algebroid cohomologies.

Extends Adams' theorem to periodic cohomology.

We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.

The paper studies cohomologies of hypercomplex manifolds and their dimensions.

We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-Kähler structure (J, ω, g). In particular, almost-Kähler manifolds satisfying a Lefschetz type property, and solvmanifolds endowed with left-invariant almost-complex structures …

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie …

Deep analysis of KV-Poisson structures and new invariant computed.

Homology and cohomology theory for topological quandles computed.

Study on deformations of Lie groupoid morphisms and their properties.

The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.

Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.

New spectral sequence for $\mathcal{K}$-manifolds, computing cohomology and harmonic forms.

This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is infinite dimensional, and also $0$- or $1$-dimensional. In the compact case, we …

Study on Riemannian foliations, decomposing cohomology based on J structure.

We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle …

Lattice cohomology, defined by Némethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two inva…

Quantum invariant derived from ternary cohomology of self-distributive structures.

We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $Σ$ and the Donaldson invariants of the algebraic surface $Σ\times P^1$. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. …

We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.

The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.

Researchers prove no unexpected relations between complex manifold numbers.

Given a smooth foliation on a closed manifold, basic forms are differential forms that can be expressed locally in terms of the transverse variables. The space of basic forms yields a differential complex, because the exterior derivative fixes this set. The basic cohomology is the cohomology of this complex, and this h…

Study complex structure deformations on Lie algebras and Dolbeault cohomology.

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

Study stabilizes Morse-Bott cohomology for equivariant manifolds.

New invariant unifies two theories of 3-manifolds, recovering quantum invariants.

Proves cohomology of elliptic structures on Lie groups can be algebraic.

In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to a $r$-simplex whose points parametrize flat connections on a smooth manifold $X$. These invariants lie in degrees $(2p-r-1)$-cohomology with $C/Z$-coefficients, for $p> r\geq 1$. In turn, this corresponds to a hom…