Algebraic K-theory of spaces, with bounded control (Q755878)

The algebraic K-theory A(X) of a space X is constructed from a category of spaces over X and homotopy equivalences between them. The author constructs a controlled analogue A(X;B), given a metric space B; it is constructed from spaces over \(X\times B\) and bounded homotopy equivalences. The main result is that the functor \(K\mapsto A(X;o(K))\) is a homology theory on piecewise linear subcomplexes of \({\mathbb{R}}^{\infty}\); here o(K) denotes an open cone on K. As a special case, it is shown that \(A(X;{\mathbb{R}}^ n)\) gives a non-connected n-fold delooping of A(X).