Surgery with Coefficients
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591 G. A. Anderson
Surgery with Coefficients I
I
Springer-Verlag Berlin. Heidelberq • New York 1977
Author Gerald A. Anderson Department of Mathematics Pennsylvania State University University Park PA 1 6 8 0 2 / U S A
AMS Subject Classifications (1970): 57 B10, 57 C10, 57 D 65 ISBN 3-540-08250-6 Springer-Verlag Berlin • Heidelberg • New York 1SBN 0-387-08250-6 Springer-Verlag New York • Heidelberg • Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg t977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCTION
This
set of notes
is d e r i v e d
at the U n i v e r s i t y
of M i c h i g a n
author's
doctoral
thesis.
complete
and
in 1973,
from a seminar and p o r t i o n s
It is i n t e n d e d
self-contained
account
to give
of surgery
given
of the
a reasonably
theory
modulo
a set of primes. The material
first
necessary
definitions exception
of relative Included
theorem
of H i r s e h
but
which
in a ring,
2 contains
Gp/H
differs
theorem.
satisfy
Normal and the
i is m a i n l y w i t h the
colocalization
immersion
the theory
is j u s t i f i e d
including
fibration.
Chapter
no new ideas,
and
of the
the b a c k g r o u n d
of
classification
and H a e f l i g e r - P o e n a r u .
collapse-expansion of spaces
the theory.
localization
The d e f i n i t i o n
and Shaneson,
contain
and contains
is a sketch
Chapter torsion.
chapters
to d e s c r i b e
and n o t a t i o n
spaces.
described
three
Chapter
3 discusses
duality
construction
invariants homotopy
modulo
groups
Whitehead
from the one given
by a W h i t e h e a d - t y p e
Poincare
the
of local
with
by Cappell local
the theory
coefficients
of a local
Spivak
a set of primes
of the
classifying
normal
are space
are computed. Chapter
theorem.
Briefly,
obstruction
4 contains groups
to f i n d i n g
the m a i n
surgery
are c o n s t r u c t e d
a homotopy
obstruction
to m e a s u r e
equivalence
the
(over a ring
and
with given torsion) dimension,
cobordant to a given map.
Below the middle
the technique is due to Milnor and Wallace.
homotopy equivalences over the integers,
~he simply connected
case is essentially done by Kervaire and Milnor, by Browder and Novikov;
Considering
and globalized
the general case is due to Wall.
We
show that the obstruction lies in a Wall group of a localized group ring. Surgery over a field was first considered by Petrie and Passman,
and Miscenko noticed that Wall's groups behaved
nicely away from
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