Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory
type: Item
instance of: scholarly article
title: Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory
zbMATH Open document ID: 0838.55004
DOI: 10.1016/0040-9383(94)00033-H
author: Erik K. Pedersen
author: Gunnar Carlsson
published in: Topology
publication date: 1996-06-03T00:00:00Z
review text: This paper deals with the Novikov conjecture in the form of the following question: For which groups \(\Gamma\) (and rings \(R)\) does the assembly map define a split injection of spectra (up to homotopy) \(B \Gamma_\wedge C(R) \to C (R \Gamma)\)? Here \(C\) is the (non-connective) Gersten-Wagoner \(K\)-theory spectrum functor, \(K^{- \infty}\), or the periodic (Wall surgery) \(L\)-theory spectrum functor, \(\mathbb{L}^{- \infty}\). It is shown that the answer to this question is positive for \(\Gamma\) provided the classifying space \(B\Gamma\) is finite and the universal free \(\Gamma\)-space \(E\Gamma\) admits a compactification \(X\) such that (i) The \(\Gamma\)-action extends over \(X\); (ii) \(X\) is metrizable; (iii) \(X\) is contractible; (iv) Compact subsets of \(E \Gamma\) become small near the boundary \(Y = X - E \Gamma\), i.e., for every \(y \in Y\), every compact set \(K \subseteq E \Gamma\), and every neighbourhood \(U\) of \(u\) in \(X\), there is a neighborhood \(V\) of \(y\) in \(X\) such that if \(g \in \Gamma\) and \(gK \cap V \neq \emptyset\), then \(gK \subseteq U\). For the \(L\)-theory case, it is further assumed that \(R\) is equipped with an involution, and that \(K_{-i} (R)\) vanishes for sufficiently large \(i\). The proofs utilize the continuously controlled \(K\)-theory of Anderson, Connolly, Ferry and Pedersen, and Ranicki's \(L\)-theory of additive categories with involution.
reviewed by: Hans Jørgen Munkholm
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 794448
zbMATH Keywords: higher signature
zbMATH Keywords: homotopy invariance
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: continuous control
Wikidata QID: Q123022483
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 794448