ALGEBRAIC K-THEORY WITH CONTINUOUS CONTROL AT INFINITY

[摘要] Let (EBAR, SIGMA) be a pair of spaces consisting of a compact Hausdorff space EBAR and a closed subspace SIGMA. Let U be an additive category. This paper introduces the category B(EBAR, SIGMA; 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(EBAR, SIGMA; U) are generalizations of the categories C(Z, U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here (Z, rho) 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; A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B(EBAR, SIGMA; U) depends only on the topology of (EBAR, SIGMA) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

[发布日期] 1994-06-03