Continuously controlled K-theory with variable coefficients Original Research Article

The purpose of this paper is to use geometric modules and path matrix morphisms of construct a continuously controlled K-theory with variable coefficients. The theory constructed here can be thought of as a “pushout” of the boundedly controlled K-theory with variable coefficients constructed in D.R. Anderson, H.J. Munkholm (Geometric modules and algebraic K-Homology theory, K-Theory 3 (1990) 561–602) and the continuously controlled K-theory with constant coefficients constructed in D.R. Anderson, F.X. Connolly, S. Ferry, E.K. Pedersen (Algebraic K-theory with continuous control at infinity, J. Pure Appl. Algebra 94 (1994) 25–47) over the boundedly controlled K-theory with constant coefficients constructed (E.K. Pedersen, C. Weibel, K-Theory Homology of Spaces, Lecture Notes in Mathematics, vol. 1370, Springer, Berlin, New York, 1989, pp. 346–361). This theory should be directly and easily applicable in the study of stratified spaces. This paper also relates the theory constructed here to the controlled K-theory constructed in D.R. Anderson, F.X. Connolly, H.J. Munkholm (A comparison of continuously controlled and controlled K-theory, Topology Appl. 71 (1996) 9–46) as an inverse limit of var epsilon-controlled K-groups and shows that, under suitable conditions, the controlled K-theory of D.R. Anderson, F.X. Connolly, H.J. Munkholm (A comparison of continuously controlled and controlled K-theory, Topology Appl. 71 (1996) 9–46) is a Quinn homology theory.