EquivariantKK-theory and the Novikov conjecture

Kasparov, Gennadi

doi:10.1007/bf01404917

Cited by 713 publications

(913 citation statements)

References 29 publications

(18 reference statements)

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“…In the late nineties, before the paper by Forrest and Hunton was written, Mihai Pimsner suggested to one of the authors 1 a method to generalize the theorem to Z d -actions. This spectral sequence was used and described already in [12] and is a special case of the Kasparov spectral sequence [40] for KKtheory. We recall it here for completeness and to justify naming H * P V after Pimsner and Voiculescu.…”

Section: Resultsmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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Section: Resultsmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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“…One could trace this through almost all of the analytic papers on the Novikov conjecture, especially in [Mis3], [HsR], [Gr], and [Hu], but to the limit the discussion I will concentrate here on the programs of Kasparov and Roe as expressed, say, in [Kas4], [Kas5], [Roe2], and [HR2]. Now let's specialize to the case where the coefficient ring R is R or C. We'll write everything out for the case of R, but C works exactly the same way.…”

Section: Theorem (Essentially Due To Karoubimentioning

confidence: 99%

“…The purpose of this note is to "explain" the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KK-approach to the Novikov conjecture (for which the key details appear in [Kas4], [Fac2], and [KS]), since this has already been done in the convenient expository references [Fac1], [Kas2], [Kas3], [Bla], and [Kas5]. Nor do I intend to explain the approach to the Novikov conjecture taken by Mishchenko and Soloviev (found in [Mis1], [Mis2], [MS], [Mis3], and [KS, Appendix]), using Fredholm representations but not using KK, for which a convenient expository reference is [HsR].…”

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confidence: 99%

Analytic Novikov for topologists

Rosenberg¹

1995

Novikov Conjectures, Index Theorems, and Rigidity

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Abstract. We explain for topologists the "dictionary" for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:(1) Why do the analytic proofs of the Novikov conjecture require the introduction of C * -algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of L-theory? Aren't they computing the wrong thing? (3) How can one show that the index map µ or β studied by operator theorists matches up with the assembly map in surgery theory? (4) Where does "bounded surgery theory" appear in the analytic proofs?Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?The literature on the Novikov conjecture (see [FRR]) consists of several different kinds of papers. Most of these fall into two classes: those based on topological arguments, usually involving surgery theory, and those based on analytic arguments, usually involving index theory. The purpose of this note is to "explain" the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KK-approach to the Novikov conjecture (for which the key details appear in [HsR]. Rather, I intend to concentrate on explaining the "dictionary" for relating the two main classes of papers on the Novikov conjecture, the topological and the analytic, and on trying

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“…A C * -bundle A(X ) over X in the sense of [6] is exactly a C 0 (X )-algebra in the sense of Kasparov [12]. That is, A(X ) is a C * -algebra together with a nondegenerate * -homomorphism [3] Parametrized strict deformation quantization of C * -algebras 27 called the structure map, where Z M(A) denotes the centre of the multiplier algebra M(A) of A.…”

Section: * -Bundles and Fibrewise Smooth * -Bundlesmentioning

confidence: 99%

PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C^-BUNDLES AND HILBERT C^-MODULES

Hannabuss

Mathai²

2011

J. Aust. Math. Soc.

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In this paper, we review the parametrized strict deformation quantization of C * -bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H 3 -twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrized strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrized strict deformation quantization of Hilbert C * -modules over C * -bundles with fibrewise torus action.2010 Mathematics subject classification: primary 58B34; secondary 81S10, 46L87, 16D90, 53D55.

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EquivariantKK-theory and the Novikov conjecture

Cited by 713 publications

References 29 publications

A spectral sequence for theK-theory of tiling spaces

A spectral sequence for theK-theory of tiling spaces

Analytic Novikov for topologists

PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C^-BUNDLES AND HILBERT C^-MODULES

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EquivariantKK-theory and the Novikov conjecture

Cited by 713 publications

References 29 publications

A spectral sequence for theK-theory of tiling spaces

A spectral sequence for theK-theory of tiling spaces

Analytic Novikov for topologists

PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C*-BUNDLES AND HILBERT C*-MODULES

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Product

Resources

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PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C^-BUNDLES AND HILBERT C^-MODULES