Bivariant Hopf Cyclic Cohomology

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories.     

Relative Group Cohomology and The Orbit Category

Let G be a finite group and ? be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ?-projective resolution for ? when ? is the family of all subgroups H ≤ G with rk H ≤ rkG ? 1. We answer this question negatively by calculating the relative group cohomology ?H*(G, 𝔽₂) where G = ?/2 × ?/2 and ? is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ?H*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ?. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E ₂ page is isomorphic to the relative group cohomology of G.     

On Stable Equivalence of Tensor Products of Algebras

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ? _A  ? ? _B S: mod(A ? _K B ^op ) → mod(E ? _K F ^op ) induces a stable equivalence between A ? _K B ^op and E ? _K F ^op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.     

Bounded differential forms,generalized Milnor–Wood inequality and an application to deformation rigidity

We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1).     

Cohomology operations and the deligne conjecture

The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

A tight colored Tverberg theorem for maps to manifolds

We prove that any continuous map of an N-dimensional simplex ΔN with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of ΔN to the same point in M: For this we have to assume that N?(r−1)(d+1), no r vertices of ΔN get the same color, and our proof needs that r is a prime. A face of ΔN is a rainbow face if all vertices have different colors.This result is an extension of our recent “new colored Tverberg theorem”, the special case of M=Rd. It is also a generalization of Volovikov?s 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov?s proof, as well as ours, works when r is a prime power.     

Stabilité de la Cohomologie LP au Voisinage de 2 pour les Espaces Localement Symétriques

We give a sufficient condition for stability around 2 of L^p cohomology of a system over locally symmetric space.     

Sharp Lp estimates for the ∂bequation on the boundaries of real ellipsoids in Cn

In this paper we prove the best possible L^p estimates for the ?-_b -equation on the boundaries of real ellipsoids in Cⁿ, and provide examples to show why they cannot be improved.