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In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group., 28 pages, in 2 languages - English and Russian

There are two approaches to the study of the cohomology of group algebras ℝ[G]: the Eilenberg-MacLane cohomology and the Hochschild cohomology. In the case of Eilenberg-MacLane cohomology, one has the classical cohomology of the classifying space BG. The Hochschild cohomology represents a more general construction, in which the so-called two-sided bimodules are considered. The Hochschild cohomology and the usual Eilenberg-MacLane cohomology are coordinated by moving from bimodules to left modules. For the Eilenberg-MacLane cohomology, in the case of a nontrivial action of the group G on the module Ml, no reasonable geometric interpretation has been known so far. The main result of this paper is devoted to an effective geometric description of the Hochschild cohomology. The key point for the new geometric description of the Hochschild cohomology is the new groupoid Gr associated with the adjoint action of the group G. The cohomology of the classifying space BGr of this groupoid with an appropriate condition for the finiteness of the support of cochains is isomorphic to the Hochschild cohomology of the algebra ℝ[G]. Hochschild homology is described in the form of homology groups of the space BGr, but without any conditions of finiteness on chains.

We give a description of the algebra of outer derivations of the group algebra of a finitely presented discrete group in terms of the Cayley complex of the groupoid of the adjoint action of the group. This problem is a smooth version of Johnson’s problem on derivations of a group algebra. We show that the algebra of outer derivations is isomorphic to the one-dimensional compactly supported cohomology group of the Cayley complex over the field of complex numbers. Bibliography: 34 titles.

A geometrical approach to the analysis of a data bank containing information on space structures of proteins is suggested. As a result of the analysis, several relationships concerning possible conformations are found, and also there is obtained a list of polypeptides whose structures differ essentially from typical concepts of the 3D-structure of proteins. Those deviations could indicate possible errors and gaps of the interpreting models in use, as well as some new unknown relations and rules.

A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem concerning the derivations of a group algebra. It is shown that the algebra of outer derivations is isomorphic to the group of the one-dimensional cohomology with compact supports of the Cayley complex over the field of complex numbers.

This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to Mackenzie [1] can be considered as a homotopy functor T LAg from the category of smooth manifolds to the transitive Lie algebroids. The functor T LAg assigns to every smooth manifold M the set T LAg(M) of all transitive algebroids with a chosen structural finite-dimensional Lie algebra g. Hence, one can construct [2, 3] a classifying space Bg such that the family of all transitive Lie algebroids with the chosen Lie algebra g over the manifold M is in one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. This enables us to describe characteristic classes of transitive Lie algebroids from the point of view of a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with those derived from the Chern-Weil type homomorphism by Kubarski [5]. As a matter of fact, we show that the Chern-Weil type homomorphism by Kubarski does not cover all characteristic classes from the categorical point of view.

Let g be a finite-dimensional Lie algebra and L be a Lie algebra bundle (LAB). A given coupling Ξ between the LAB L and the tangent bundle TM of a manifold M generates the so-called Mackenzie obstruction Obs(Ξ) ∈ H 3 (M; ZL) to the existence of a transitive Lie algebroid (K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, 2005, p. 279). We present two cases of evaluating the Mackenzie obstruction. In the case of a commutative algebra g, the group Aut(g) δ is isomorphic to the group of all matrices GL(g) with the discrete topology. We show that the Mackenzie obstruction for coupling Obs(Ξ) vanishes. The other case describes the Mackenzie obstruction for simply connected manifolds. We prove that, for simply connected manifolds, the Mackenzie obstruction is also trivial, i.e. Obs(Ξ) = 0 ∈ H 3(M; ZL; ∇ Z ).

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝn to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus \(\mathbb{T}^{2n} \). This reduction enables us to identify the Schwartz space on ℝn with the space of smooth sections Γ∞(T2n, ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ℝn, we construct an index formula in accordance with the classical Atya-Singer formula.

We investigate orthonormality-preserving, C ⁎ -conformal and conformal module mappings on full Hilbert C ⁎ -modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C ⁎ -algebra of coefficients combined with an isometric module operator as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfills the equality 〈 T ( x ) , T ( y ) 〉 = | λ | 2 〈 x , y 〉 for any elements x, y of the Hilbert C ⁎ -module. At the contrary, C ⁎ -conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators.

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ((10), 1972) and Gromov ((4), 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In (6), we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

The signature of the Poincare duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V ⊕ V ∗ where V is abstract linear space with countable base. The space W is considered with minimal natural topology. The symmetric quadratic form on the space W is generated by the Poincare duality homomorphism on the abstract chain–cochain groups induced by singular simplices on the topological manifold. © 2008 Elsevier B.V. All rights reserved. MSC: 19Lxx; 55N25; 57Nxx

We present several geometrical and statistical methods analyzing information contained in Protein Data Bank for consistency and general theory fitting. In addition, we include a list of examples demonstrating the necessity of careful analysis and sampling of the PDB items. In particular, we show some examples visualizing the distribution of disturbance level of the famous Pauling plane low. Besides, we present a method of automatical helices finding based on discrete analogues of curvature and torsion of smooth curve.

It is proved that for any transitive Lie algebroid on a compact oriented connected manifold with unimodular isotropy Lie algebras and trivial monodromy the cohomology algebra is a Poincare algebra with trivial signature. Examples of such algebroids are algebroids on simply connected manifolds, algebroids such that the outer automorphism group of the isotropy Lie algebra is equal to its inner automorphism group, or such that the adjoint Lie algebra bundle induces a trivial homology bundle in the category of flat bundles.

An almost representation of a group is a map from this group into the unitary group of a Hilbert space, such that the group relations hold only approximately. We give a survey of the recent results on almost representations and on their relations to asymptotic representations. Applications to K-theory of classifying spaces are also discussed.

We prove a formula that expresses the square of the statistical sum for the two-dimensional Ising model for an arbitrary plane lattice in terms of the determinant of a matrix similar to the Kac-Ward matrix.

In this paper, we present an analytic definition for the relative torsion for flat C∗-algebra bundles over a compact manifold. The advantage of such a relative torsion is that it is defined without any hypotheses on the flat C∗-algebra bundle. In the case where the flat C∗-algebra bundle is of determinant class, we relate it easily to the L2 torsion as defined in [7], [5].

A -algebra servicing the theory of asymptotic representations and its embedding into the Calkin algebra that induces an isomorphism of -groups is constructed. As a consequence, it is shown that all vector bundles over the classifying space that can be obtained by means of asymptotic representations of a discrete group can also be obtained by means of representations of the group into the Calkin algebra. A generalization of the concept of Fredholm representation is also suggested, and it is shown that an asymptotic representation can be regarded as an asymptotic Fredholm representation.

This paper precludes the proceedings of the conference “Geometry and Operator Theory” dedicated to N. Teleman’s 65th birthday celebration (Ancona, September 2007)

The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula. Earlier a natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona when the completion is a closed manifold. Here we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with boundary. The results were developed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration the case of manifolds with boundary., 10 pages. The second version is supplemented by details of consideration the case of manifolds with boundary

If one has a unitary representation p : π →U(H) of the fundamental group π1(M) of the manifold M, then one can do many useful things: 1) construct a natural vector bundle over M; 2) construct the cohomology groups with respect to the local system of coefficients; 3) construct the signature of manifold M with respect to the local system of coefficients

In the section 1.2 of the chapter 1 (Theorem 3) it was shown that if the base of a locally trivial bundle with a Lie group as structure group has the form B x I then the restrictions of the bundle to B x {0} and B x {1} are isomorphic. This property of locally trivial bundles allows us to describe bundles in the terms of homotopy properties of topological spaces.

To extend the study of the properties of the vector bundles we need further geometric ideas and constructions. This chapter is devoted to the most frequently used constructions which lead to deeper properties of vector bundles. They are Bott periodicity — the main instrument of the calculation of K-theory, linear representations and cohomology operations in K-theory and the Atiyah-Singer formula for calculating of the indices of elliptic operators on compact manifolds. The last of these will be considered in the next chapter.

In this chapter we describe some of the most fruitful applications of vector bundles, namely, in elliptic operator theory. We study some of the geometrical constructions which appear naturally in the analysis of differential and pseudodifferential operators on smooth manifolds.

The definition of a locally trivial bundle was coined to capture an idea which recurs in a number of different geometric situations. We commence by giving a number of examples.

In this chapter we shall give some methods for describing the K-groups of various concrete spaces by reducing them to a description in terms of the usual cohomology groups of spaces. The methods we discuss are spectral sequences, cohomology operations and direct images. These methods cannot be used universally but they allow us to use certain geometric properties of our concrete topological spaces and manifolds.

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [ Conjecture de Novikov et fibres presque plats , C. R. Acad. Sci. Paris Ser. I 310 (1990), 273–277]. Using a natural construction of [B. Hanke and T. Schick, Enlargeability and index theory , preprint, 2004], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles over arbitrary CW-spaces using quasi-connections [N. Teleman, Distance function, Linear quasi-connections and Chern character , IHES/M/04/27]. Connes, Gromov and Moscovici [ Conjecture de Novikov et fibres presque plats , C. R. Acad. Sci. Paris Ser. I 310 (1990), 273–277] showed that for any almost flat bundle $\alpha$ over the manifold $M$, the index of the signature operator with values in $\alpha$ is a homotopy equivalence invariant of $M$. From here it follows that a certain integer multiple $n$ of the bundle $\alpha$ comes from the classifying space $B\pi_{1}(M)$. The geometric arguments discussed in this paper allow us to show that the bundle $\alpha$ itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace $Y\subset B\pi_{1}(M)$ trough the classifying mapping.

A solution of the linear differential game of pursuit without discrimination of the evading object is constructed under certain natural constraints on the form of the differential game, such as smoothness of the boundaries of the sets of admissible values of the control parameters and conditions of general position. A solution is constructed on the basis of a choice of the pursuit control which maximizes the rate of decay of the estimator function of the time of termination of the game and is best possible in a certain sense. The asymptotic behavior of the estimator function at singular points of the differential game is investigated.Bibliography: 6 titles.

We study the admissible sets of fixed submanifolds under the action of the group Zp in the quasi-complex manifold M2n, n < p.

It is proved that if, on a manifold with an involution, the subset of fixed points is diffeomorphic to an even-dimensional real projective space, then the manifold is bordant to the complex projective space in the class of nonoriented bordisms.