A Geometric Construction of the Boundedly Controlled Whitehead Group

In our paper “An Introduction to Boundedly Controlled Simple Homotopy Theory” which appears elsewhere in these Proceedings, we introduced the notions of a space Z with a boundedness control structure (P,C) and the category of finite boundedly controlled (or, simply, bc) CW complexes over Z. This category has objects (X,p) where X is a finite dimensional CW complex and p : X → Z is a proper map with some additional properties (cf. [2; section 1]) and is denoted by CW _ _ f c / Z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq128.tif"/> . Since our paper [2] mainly surveys the development of simple homotopy theory in CW _ _ f c / Z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq129.tif"/> , it contains very few details. It is the purpose of this note to fill in some of those details by describing a construction of Wh^c(X,p), the boundedly controlled Whitehead group of (X,p), and showing that the correspondence (X,p) ↦ Wh^c(X,p) defines a homotopy functor Wh c   :   CW _ _ f c / Z   →   Ab _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq130.tif"/> , where Ab _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq131.tif"/> is the category of abelian groups. This result is Theorem 2.3 below.