Several recent investigations have focused attention on spaces and
manifolds which are non-compact but where the problems studied have some
kind of "control near infinity". This monograph introduces the category
of spaces that are "boundedly controlled" over the (usually non-compact)
metric space Z. It sets out to develop the algebraic and geometric tools
needed to formulate and to prove boundedly controlled analogues of many
of the standard results of algebraic topology and simple homotopy
theory. One of the themes of the book is to show that in many cases the
proof of a standard result can be easily adapted to prove the boundedly
controlled analogue and to provide the details, often omitted in other
treatments, of this adaptation. For this reason, the book does not
require of the reader an extensive background. In the last chapter it is
shown that special cases of the boundedly controlled Whitehead group are
strongly related to lower K-theoretic groups, and the boundedly
controlled theory is compared to Siebenmann's proper simple homotopy
theory when Z = IR or IR2.