arXiv research

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…

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

Uniform volume estimate for Kähler metrics in big cohomology classes.

Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.

Paper computes rational cohomology of spin hyperelliptic mapping class groups.

Study mapping class group action on cohomology of SL_n character variety.

Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.

We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in $R^n, n \ge 3,$ generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in $R^3$. As the first applications we give such formulas for the (r…

Paper constructs a cohomology class related to McDuff's secondary class, proving it transgresses to the Euler class of foliated sphere bundles.

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bo…

We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental group. In degree two we clarify the relation between hyperbolic and atoroidal cla…

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces,…

Injective map from top cohomology of moduli spaces to handlebodies.

In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the L…

We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in…

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…

Cohomology fractals illustrate complex 3-manifold properties.

Estimates Kaehler metrics' diameter in big cohomology classes.

Study convexity of Mabuchi functional in big cohomology classes.

Develops relative cohomology for Lie groupoids and algebroids.

The paper constructs cohomology classes on curve strata.

Novikov conjecture reduced to Lipschitz cohomology of groups.

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

Study cohomology classes related to harmonic maps on submersions.

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…

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The ma…

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

New algebraic tools solve Poisson and Lie bialgebra problems.

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection be…

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

Study on cohomology of spin hyperelliptic mapping class groups.

We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds…

New classes defined for manifold pseudogroups, linking to cohomology and bundle structures.

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…

Topological Pontryagin classes are algebraically independent in high-dimensional spaces.

Generalised characteristic classes are constructed for bordism cohomologies which allow a natural extension of classical genera to these bordism cohomology rings taking values in singular cohomology.

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.

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

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain sub…

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

Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.

Modern differential cohomology explained with applications.

Elliptic bouquets defined for spin manifolds with circular actions.

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Simil…

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $θ$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[θ]$ of the odd cocycle $…

We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated dif…

In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.

I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class gr…