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.
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