Controlled Simple Homotopy Theory and Applications
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1009
lA.Chapman
Controlled Simple Homotopy Theory and Applications
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
T. A. Chapman University of Kentucky, College of Art and Sciences Dept of Mathematics, Patterson Office Tower Lexington, KY 40506, USA
AMS-Subject Classifications (1980): 57 A 15,57 A 30 ISBN 3-540-12338-5 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12338-5 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CON TEN T S
0 Introduction
1
1 Applications
5
2 Definitions and Notation
11
3 Construction of Wh
13
4 Functorial Properties
17
5 Controlled Whitehead Torsion
19
6 Construction of Ko(Y)g
26
7 Controlled Finiteness Obstruction
28
8 Further Properties of the Controlled Finiteness Obstruction
33
9 The Splitting Homomorphism
41
10 The Splitting Sequence
50
11 The Realization Theorem
60
12 Calculations
70
13 The Controlled Boundary Theorem
79
14 The Controlled s-Cobordism Theorem
84
References
91
Index
93
CONTROLLED SIMPLE HOMOTOPY THEORY AND APPLICATIONS
O.
INTRODUCTION We start by giving a brief description of simple homotopy theory from
the classical point of view.
Simple homotopy theory is merely the obstruc-
tion theory that arises when one attempts to answer the following question: If f : X
> Y
is a /'l(Jmotopy equioal.enoe betiaeen compact polyhedra, when is f
simple?
That is, when do there exist a compact polyhedron Z and PL (= piece-
wise linear) 4urjections r : Z
>
for which for is homotopic to s?
X,
S
Z > Y having contractible pointinverses
Of course the usual formulation of this
definition requires rand s to be collapses, but it is wellknown that our formulation using CEPL maps (= PL surjections with contractible pointinverses) is equivalent to the standard one [8, Theorem 11.1].
To answer the above
question one first defines the Whitehead group of Y, Wh(Y) , which is an abelian group that may be defined algebraically in terms of the fundamental group of Y [9, p.39], or geometrically [9, p.20].
Then the above question is
answered by assigning to each homotopy equivalence f : X > Y a unique (Whitehead) torsion L(f) E Whey) which is 0 if and only if f is simple [9, p.22]. The above construction takes on some added significance in light of some wellknown useful calculations.
For example, if the fundamental group of Y is
trivial, then Wh(Y)=0 and so f is always simple [9, p.32].
A much deeper cal-
culation is the result of Bass
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