Algebraic K-theory with continuous control at infinity

Abstract

Let ( E ̄ , Σ ) be a pair of spaces consisting of a compact Hausdorff space Ē and a closed subspace Σ. Let U be an additive category. This paper introduces the category B( E ̄ , Σ; U of geometric modules over E with coefficients in U and with continuous control at infinity. One of the main results is to show that the functor that sends a CW complex X to the algebraic K-theory of B(cX, X; U) is a homology theory. Here cX is the closed cone on X and X is its base. The categories B( E ̄ , Σ; U) are generalizations of the categories C(Z; U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here ( Z, ϱ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C( O(X); U)→ B(cX, X; U) . A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B( E ̄ , Σ; U) depends only on the topology of ( E ̄ , Σ ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.