Algebraic \(K\)-theory with continuous control at infinity
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory with continuous control at infinity
zbMATH Open document ID: 0854.55003
DOI: 10.1016/0022-4049(94)90004-3
author: Douglas R. Anderson
author: Francis Connolly
author: Steve Ferry
author: Erik K. Pedersen
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-19T00:00:00Z
review text: In controlled geometry, constructions are parametrized by a space \(E\), and many obstructions lie in some \(K\)-theoretic group associated to \(E\). Until this paper, \(E\) had to be a metric space in order to say that operations (or geometric modules) were appropriately ``bounded''. The paper under review only supposes that \(E\) is an open subspace of a compact Hausdorff space \(\overline{E}\), and introduces the new notion of geometric modules with continuous control at infinity. The \(K\)-theory \(K_* {\mathcal B}\) of the category \({\mathcal B} = {\mathcal B}(\overline{E}, \overline{E} - \overline{E})\) of such modules is the main focus of the article. Given these notions, the authors proceed to compare their construction to the bounded theory in [\textit{E. K. Pedersen} and the reviewer, Lect. Notes Math. 1370, 346-361 (1989; Zbl 0674.55006)]. When \(E\) is the open cone of a compact metrizable space \(X\), they show that \(H_* (X) = K_* {\mathcal B}\) is a generalized homology theory, and that it agrees with the homology theory of [loc. cit.] when \(X\) is a finite CW complex.
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 19D06
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 609789
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: homology theory
zbMATH Keywords: controlled geometry
zbMATH Keywords: continuous control
zbMATH Keywords: \(K\)-theory
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90004-3
OpenAlex ID: W2082152552
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description: scientific article; zbMATH DE number 609789