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Cyclic cohomology, the Novikov conjecture and hyperbolic groups

Cyclic cohomology, the Novikov conjecture and hyperbolic groups,10.1016/0040-9383(90)90003-3,Topology,A Connes

Cyclic cohomology, the Novikov conjecture and hyperbolic groups   (Citations: 156)
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Journal: Topology , vol. 29, no. 3, pp. 345-388, 1990
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    • ...Let D be an elliptic differential operator on a compact manifold M. As is wellknown ellipticity implies that D is a Fredholm operator and the Atiyah-Singer index theorem [ATSI] expresses the index of D as a topological formula involving the Chern character of the symbol s(D) and the Todd class of the manifold M. In [COMO], CONNES–MOSCOVICI proved a far reaching generalization of the Atiyah–Singer index theorem, the so-called higher index ...
    • ...the pairing ind[f ](D) = h X[f ],Ch(eD)i . The higher index theorem in [COMO] computes this number - which no longer is integral - in terms of topological data generalizing the Atiyah–Singer index theorem...
    • ...rive CONNES–MOSCOVICI’s higher index from the higher algebraic index theorem, we prove that the algebraic pairing h Q(a), P1 − P2i coincides asymptotically with the pairing h X[f ],Ch(eD)i defined in [COMO]...
    • ...Recall that CONNES–MOSCOVICI [COMO] used their higher index theorem to prove a covering index theorem, which was used to prove the Novikov conjecture...
    • ...CONNES–MOSCOVICI defined in [COMO, §2.]...
    • ...First, the map ¶ : K1 C ¥(S∗Q) � → K0 YDO−¥(Q) � is the index map (cf. [COMO,...
    • ...Let us explain, why the definition of the localized index by Eq. 6.19 coincides with the original definition by [COMO]...
    • ...[COMO, p. 353], and that this operator is homotopic to the graph projection of D (cf...
    • ...Hereby,V1 −V2 is the virtual vector bundle obtained by the asymptotic limit ¯h ց 0 of r. We have thus reproved the following result from [COMO]...

    M. J. Pflaumet al. Cyclic cocycles on deformation quantizations and higher index theorems

    • ...and one wishes to calculate the diagonal map �. From cyclic cohomology invariants of B one thus gets K-theory invariants of A ; this is used for instance in the formulation of higher index theorems [6]...

    Denis Perrot. Quasihomomorphisms and the residue Chern character

    • ...For the conjugacy class of the unit this result is due to Connes and Moscovici [CM] and plays a crucial role in their proof of the Novikov higher signature conjecture for word-hyperbolic groups...
    • ...nents. On the other hand the closure of the subcomplex CChei( l C) corresponding to the conjugacy class of the unit element denes a direct topological summand of the analytic cyclic bicomplex of any unconditional admissible Fr echet algebra over l C. This was observed by Connes and Moscovici in [CM] and plays a crucial role in their proof of the Novikov higher signature conjecture for word-hyperbolic groups...
    • ...Proof: Using the ideas of [CM], this is an immediate consequence of 6.3...

    Michael Puschnigg. New Holomorphically Closed Subalgebras of C* -Algebras of Hyperbolic G...

    • ...In [40] D. Tamarkin and B. Tsygan, inspired by the Connes-Moscovici higher index formulas [14], suggested the rst version of the algebraic index theorem for a Poisson manifold...

    V. A. Dolgushevet al. An algebraic index theorem for Poisson manifolds

    • ...One of the main motivations to study the index theory of Dirac operator twisted by C ∗-vector bundles comes from higher index theory, pioneered by Connes-Moscovici with their higher index theorem and relevant in connection with the Novikov conjecture [CM]...
    • ...r. An appropriate system of Banach algebras can be gained fro m the work of Connes-Moscovici [CM]...
    • ...In the following let A = C ∗. We assume that the projective system ( Ai)i∈IN0 is such that C ⊂ A∞. For the reduced group C ∗-algebra C ∗ r a suitable projective system of algebras (Bi)i∈IN can be derived from a construction of Connes–Moscovici [CM], see for example [W2, §4] for a detailed account...

    Charlotte Wahl. The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-alg...

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