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