A Geometric Construction of the Boundedly Controlled Whitehead Group

Abstract

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 Whc(X,p), the boundedly controlled Whitehead group of (X,p), and showing that the correspondence (X,p) ↦ Whc(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.