Hopf-cyclic homology and cohomology with coefficients

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Cup products in Grassmannians

This note calculates the height of the first Stiefel-Whitney class in the cohomology of the real Grassmannians and determines the length of the longest nontrivial cup-product in $\text{H}{}^1\text{(G}{}_{\mathrm{k}}\text{(R}{}^{\mathrm{n+k}}\text{);Z}{}_2\text{) (k?n)}$ with k?4.     

Measurable Kac cohomology for bicrossed products

We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a measurable version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.

     

Noncommutative Mackey functors and Hopf-cyclic Homology

We show that a duality of the Hopf-cyclic homology and cohomology can be explained in terms of functors defined on a PROP for Hopf algebras.     

Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

Cup and cap products in intersection (co)homology

We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincaré duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.     

On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.     

Künneth formulae and cross products for the symplectic Floer cohomology

For a symplectic monotone manifold (P,ω) and φSymp₀(P,ω), we define a -graded symplectic Floer cohomology (a local invariant) over integral coefficients. There is a spectral sequence which arises from a filtration on the -graded symplectic Floer cochain complex. The spectral sequence converges to the -graded symplectic Floer cohomology (a global invariant). We show that there are cross products on the -graded symplectic Floer cohomology and on the spectral sequence, hence on the usual -graded symplectic Floer cohomology. The Künneth formula for the -graded symplectic Floer cohomology is proved and similar results for the spectral sequence are obtained.     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

Cup products,lower central series,and holonomy Lie algebras

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra of a group, from finitely presented, commutator-relators groups to arbitrary finitely presented groups. Using the notion of “echelon presentation,” we give an explicit formula for the cup-product in the cohomology of a finite 2-complex. This yields an algorithm for computing the corresponding holonomy Lie algebra, based on a Magnus expansion method. As an application, we discuss issues of graded-formality, filtered-formality, 1-formality, and mildness. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.