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Register a new userMod-2 Equivalence of the K-theoretic Euler and Signature Classes
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This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.
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This paper deals with two aspects of the theory of characteristic classes of star products: first, on an arbitrary Poisson manifold, we describe Morita equivalent star products in terms of their Kontsevich classes; second, on symplectic manifolds, we describe the relationship between Kontsevichs and Fedosovs characteristic classes of star products.
Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $phi, psi: Cto A$ are approximately unitarily equivalent if and only if $$ [phi]=[psi] {rm in} KL(C,A) and taucirc phi=taucirc psi tforal tauin T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $kappain KL(C,A)$ with $kappa(K_0(C)_+setminus {0})subset K_0(A)_+setminus {0}$ and with $kappa([1_C])=[1_A]$ and a continuous affine map $lambda: T(A)to T_{mathtt{f}}(C)$ which is compatible with $kappa,$ where $T_{mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $phi: Cto A$ such that $$ [phi]=kappaandeqn taucirc phi(c)=lambda(tau)(c) $$ for all $cin C_{s.a.}$ and $tauin T(A).$ Denote by ${rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ Lambda: {rm Mon}_{au}^e (C,A)to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $kappain KL(C(X), A)$ with $kappa(K_0(C(X))_+setminus{0})subset K_0(A)_+setminus {0}$ for which there is no hm $h: C(X)to A$ so that $[h]=kappa.$
We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational pm (2,2)-moves (i.e. pm 5/2-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2,2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.
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