In the present paper we study some homotopy invariants which can be defined by means of bundles with fiber being a matrix algebra. In particular, we introduce some generalization of the Brauer group in the topological context and show that any of its elements can be represented as a locally trivial bundle with the structure group , k ∈ . Finally, we discuss its possible applications in the twisted K-theory.

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

The algebra Ψ(M) of order zero pseudodifferential operators on a compact manifold M defines a well-known C*-extension of the algebra C(S*M) of continuous functions on the cospherical bundle S*M ⊂ T*M by the algebra К of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism T from C0(T*M) to К, which plays the role of a deformation for the commutative algebra C0(T*M). Similar constructions exist also for operators and symbols with coefficients in a C*-algebra. Recently we have shown that the image of the above extension under the Connes–Higson construction is T and that this extension can be reconstructed out of T. That is why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms. But the image of the above extension is defined only outside the zero section of T*(M), so it may seem that the information encoded in the extension is not the same as that in the asymptotic homomorphism. We show that this is not the case.

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.

Starting from the candidate Bloch-Beilinson filtration on constructed in [L2], we develop and describe geometrically a series of Hodgetheoretic invariants Ψi defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the Ψi, with applications to 0-cycles on products of curves.

It is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.

Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.

In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.

The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

C*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

In this paper we consider reduction of nontorsion elements in the étale and Quillen K-theory of a curve X over a number field. As an application we solve two problems: detecting linear dependence and the support problem.

We construct dihedral and reflexive cohomology theories for *-Hopf algebras. This generalizes the Connes–Moscovici construction of cyclic cohomology for Hopf algebras.

Bases are constructed for the homogeneous components of the invariant subalgebra and anti-invariant submodule of the algebra K*K(p) of cooperations for p-local complex K-theory.

The purpose of the present mostly expository paper (based mainly on [17, 18, 40, 16, 11]) is to present the current state of the following conjecture of A. Fel'shtyn and R. Hill [13], which is a generalization of the classical Burnside theorem.

Let G be a countable discrete group, φ one of its automorphisms, R(φ) the number of φ-conjugacy (or twisted conjugacy) classes, and S(φ) = #Fix the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.

This conjecture plays a important role in the theory of twisted conjugacy classes (see [26], [10]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative Harmonic Analysis.

First we prove this conjecture for finitely generated groups of type I and discuss its applications.

After that we discuss an important example of an automorphism of a type II1 group which disproves the original formulation of the conjecture.

Then we prove a version of the conjecture for a wide class of groups, including almost polycyclic groups (in particular, finitely generated groups of polynomial growth). In this formulation the role of an appropriate dual object plays the finite-dimensional part of the unitary dual. Some counter-examples are discussed.

Then we begin a discussion of the general case (which also needs new definition of the dual object) and prove the weak twisted Burnside theorem for general countable discrete groups. For this purpose we prove a noncommutative version of Riesz-Markov-Kakutani representation theorem.

Finally we explain why the Reidemeister numbers are always infinite for Baumslag-Solitar groups.

In the present note we prove formality of the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold.

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

Let G denote a finite group and n = 2k 6 an even integer. Let X denote a simply connected, compact, oriented, smooth G-manifold of dimension n. Let L denote a union of connected, compact, neat submanifolds in X of dimension k. We invoke the hypothesis that L is a G-subcomplex of a G-equivariant smooth triangulation of X and contains the singular set of the action of G on X. If the dimension of the G-singular set is also k then the ordinary equivariant self-intersection form is not well defined, although the equivariant intersection form is well defined. The first goal of the paper is to eliminate the deficiency above by constructing a new, well defined, equivariant, self-intersection form, called the generalized (or doubly parametrized) equivariant self-intersection form. Its value at a given element agrees with that of the ordinary equivariant self-intersection form when the latter value is well defined. Let denote a finite family of immersions withtrivial normal bundle of k-dimensional, connected, closed, orientable, smooth manifolds into X. Assume that the integral (and mod 2) intersection forms applied to members of and to orientable (and nonorientable) k-dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on × and the generalized equivariant self-intersection form on is a necessary and sufficient condition that is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L. This property, when is a finite family of immersions of the k-dimensional sphere Sk into X, is just what is needed for constructing an equivariant surgery theory for G-manifolds X as above whose G-singular set has dimension less than or equal to k. What is new for surgery theory is that the equivariant surgery obstruction is defined for an almost arbitrary singular set of dimension k and in particular, the k-dimensional components of the singular set can be nonorientable.

The paper introduces a group LSP of obstructions to splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the LSP-groups with various surgery obstruction groups for a manifold triple and structure sets arising from a manifold triple. The natural map from the surgery obstruction group of the ambient manifold to the LSP-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some LSP-groups are computed precisely.

Given a suitable functor T : → between model categories, we define a long exact sequence relating the homotopy groups of any X ε with those of TX, and use this to describe an obstruction theory for lifting an object G ε to . Examples include finding spaces with given homology or homotopy groups.