Alexander S. Mishchenko

Asymptotic Representation of Discrete Groups

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;

Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism

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.

Mackenzie obstruction for the existence of a transitive Lie algebroid

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) ∈ H3 (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 ∈ H3(M; ZL; ∇Z).

Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

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.

Almost flat bundles and almost flat structures

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

Critical Analysis of Amino Acids and Polypeptides Geometry

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Construction of Fredholm Representations and a Modification of the Higson-Roe Corona

over the manifold

Collection of materials presented at the conference “Geometry and Operator Theory” dedicated to N. Teleman’s 65th birthday celebration, Ancona, 20–22 September 2007

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Geometric Constructions of Bundles

, the index of the signature operator with values in

Introduction to the Locally Trivial Bundles Theory

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