Equivariant KK-theory and the Novikov conjecture
type: Item
instance of: scholarly article
title: Equivariant KK-theory and the Novikov conjecture
zbMATH Open document ID: 0647.46053
DOI: 10.1007/BF01404917
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143536
review text: This fundamental paper is the definitive version with complete proofs of a ``conspectus'' containing a sketch of the proofs circulated since 1981. The author constructs a G-equivariant bivariant K-theory KK G(A,B) where the arguments are C *-algebras with a continuous G-action and G is a locally compact (not necessarily compactö) group. This theory is then applied to the Novikov conjecture. The author proves a result which contains and considerably strengthens all previously known results on this conjecture. As in ordinary Kasparov theory the main technical tool is the intersection product KK G(A,B)\(\times KK\) G(B,C)\(\to KK\) G(A,C) and so called Dirac- and dual Dirac elements where the first one comes from a Dirac operator and the second one is (in good cases) a right inverse. The author's approach actually proves stronger versions of the Novikov conjecture (which imply the ordinary conjecture) for a large class of fundamental groups \(\pi\), for instance the following conjecture \(SNC_{\beta}:\) The natural homomorphism \(\beta\) from the representable K-homology of the classifying space \(B\pi\) to the K-theory of the group-C *-algebra C *(\(\pi)\) is split injective. Here is then a sample result: Theorem 6.7. Suppose that \(\pi\) is a discrete group such that the universal covering space \(X=E\pi\) of the classifying space \(B\pi\) is a special \(\pi\)-manifold (i.e. admits a ``dual Dirac'' element). Then \(SNC_{\beta}\) is true for \(\pi\). In particular, \(SNC_{\beta}\) is true for all groups \(\pi\) for which \(B\pi\) can be taken as a complete Riemannian manifold of non-positive sectional curvature, and also for all closed, discrete, torsionless subgroups of finite component Lie groups. Independently of its applications to the Novikov conjecture, the equivariant KK G-theory developed in this paper is a powerful tool and an important achievement by itself. It allows, for instance, to mention only one thing, to define a ``topological representation ring'' \…
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4056334
zbMATH Keywords: G-equivariant bivariant K-theory
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: Kasparov theory
zbMATH Keywords: dual Dirac elements
zbMATH Keywords: Dirac operator
zbMATH Keywords: fundamental groups
zbMATH Keywords: representable K- homology
zbMATH Keywords: classifying space
zbMATH Keywords: split injective
zbMATH Keywords: torsionless subgroups
zbMATH Keywords: topological representation ring
zbMATH Keywords: crossed product C *-algebras
author: Gennadi Kasparov
reviewed by: Joachim Cuntz
MaRDI profile type: Publication
Wikidata QID: Q123357542
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description: scientific article; zbMATH DE number 4056334