Algebraic \(K\)-theory with continuous control at infinity
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.
scientific article; zbMATH DE number 609789
scientific article