The integral \(K\)-theoretic Novikov conjecture for groups with finite asymptotic dimension
type: Item
instance of: scholarly article
title: The integral \(K\)-theoretic Novikov conjecture for groups with finite asymptotic dimension
zbMATH Open document ID: 1071.19003
author: Gunnar Carlsson
author: Boris Goldfarb
published in: Inventiones Mathematicae
publication date: 2005-01-25T00:00:00Z
review text: This selfcontained paper is devoted to provide a proof of a result closely related to the integral \(K\)-theoretic Novikov conjecture for the case of groups with finite asymptotic dimension, using techniques developed by \textit{G. Carlsson} [``Bounded \(K\)-theory and the assembly map in algebraic \(K\)-theory'', in: Novikov conjectures, index theory and rigidity. Vol. 2, Lond. Math. Soc. Lect. Note Ser. 227, 5--127 (1995; Zbl 0957.19001)]. As defined by \textit{M. Gromov} [``Asymptotic invariants of finite groups'', in: Geometric group theory. Vol. 2, Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)], a pair of subspaces of a given metric space \(X\) are said to be \(d\)-disjoint if the distance between then is larger than \(d\). The asymptotic dimension \(X\) is defined to be the smallest number \(n\) such that there is a uniformly bounded cover \({\mathcal U}\) of \(X\) by \(n+1\) \(d\)-disjoint families of subsets \(U=U_1,\ldots,U_n\). Given a discrete group \(\Gamma\) and a ring \(R\), an element \(\gamma\in\Gamma\) may be interpreted as an isomorphism of the trivial \(R[\Gamma]\)-module, and to each isomorphism \(f\) of finitely generated \(R[\Gamma]\)-modules corresponds an isomorphism \(\gamma\otimes f\) of finitely generated \(R[\Gamma]\)-modules. From the main result in this paper it follows that this is a functor which induces an assembly map \(\alpha:B\Gamma_+\wedge K(R)\rightarrow K(R[\Gamma])\). Here \(B\Gamma\) is the compact universal space of the group \(\Gamma\), the map \(\alpha:B\Gamma_+\wedge K(R)\) is the nonconnective \(K\)-theory spectrum of the group \(\Gamma\) with coefficients in the nonconnective \(K\)-theory spectrum of \(R\) and \(K(R[\Gamma])\) is the nonconnective \(K\)-theory spectrum of the group ring \(R[\Gamma]\). The integral Novikov conjecture predicts that this map is a split injection for any \(R\) and any group \(\Gamma\) with \(B\Gamma\) being a finite \(CW\)-complex. The main result is the following theorem: Let…
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 2130987
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: asymptotic dimension
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
full work available at URL: s00222-003-0356-x
OpenAlex ID: W2002035035
Wikidata QID: Q123222681
cites work: On asymptotic dimension of groups
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cites work: The Novikov conjecture for groups with finite asymptotic dimension
DOI: 10.1007/S00222-003-0356-X
description: scientific article; zbMATH DE number 2130987