Surgery with Coefficients

164 Pages·1977·4.872 MB·English

by  G. A Anderson

Lecture Notes ni
Mathematics
Edited yb .A Dold and .B Eckmann
195
G. A. Anderson
Surgery with Coefficients
I I
Springer-Verlag
Berlin. Heidelberq • New York 91 77
Author
Gerald A. Anderson
Department of Mathematics
Pennsylvania State University
University Park
PA 16802/USA
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
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INTRODUCTION
This set of notes is derived from a seminar given
at the University of Michigan in 1973, and portions of the
author's doctoral thesis. It is intended to give a reasonably
complete and self-contained account of surgery theory modulo
a set of primes.
The first three chapters contain the background
material necessary to describe the theory. Chapter i is mainly
definitions and notation and contains no new ideas, with the
exception of relative localization and colocalization of
spaces. Included is a sketch of the immersion classification
theorem of Hirseh and Haefliger-Poenaru.
Chapter 2 contains the theory of local Whitehead
torsion. The definition differs from the one given by Cappell
and Shaneson, but is justified by a Whitehead-type local
collapse-expansion theorem. Chapter 3 discusses the theory
of spaces which satisfy Poincare duality with coefficients
in a ring, including the construction of a local Spivak normal
fibration. Normal invariants modulo a set of primes are
described and the homotopy groups of the classifying space
Gp/H are computed.
Chapter 4 contains the main surgery obstruction
theorem. Briefly, groups are constructed to measure the
obstruction to finding a homotopy equivalence (over a ring and
with given torsion) cobordant to a given map. Below the middle
dimension, the technique is due to Milnor and Wallace. Considering
homotopy equivalences over the integers, ~he simply connected
case is essentially done by Kervaire and Milnor, and globalized
by Browder and Novikov; 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 the prime .2 More recently, Connelly, Giest
and Pardon have considered rational surgery (in the non-simple
case), and the methods of Cappell and Shaneson (which uses
rings with a local epimorphism ~ + R) also apply. The
general case, with rings of the form R~, is due to the author
in his thesis.
Chapter 5 gives the geometric definition of surgery
groups, and the generalization to manifold n-ads. Quinn's
approach is also briefly discussed. Finally, the periodicity
theorem, in the non-simple case, is proved.
Chapter 6 describes the result of changing rings
in surgery groups by means of a long exact sequence. Corollaries
include a Rothenberg-type sequence, the general periodicity
isomorphism and determination of the kernel of
s ~) s
L2k_l( ÷ L2k_l(~) ~ finite, by simple linking forms,
generalizing the original odd-dimensional surgery obstructions
due to Wall and clarified by Connelly.
Finally, five appendicies are included: Whitehead
torsion notions for n-ads, the algebraic construction of
Ln(W÷w';R) , the computation of Ln(~;R), surgery on embedded
manifolds, and homotopy and homology spheres. The has reference been
arranged into categories. Undoubtedly, some errors and
omissions have occurred in this arrangement, but I hope the
general drift is helpful to the reader.
A number of people have been of greaD help in
writing these notes. I am indebted to my thesis advisor
C.N. Lee for many helpful suggestions and discussions. I
would also llke to thank Dennis Barden, Allan Edmonds,
Steve Ferry, and Steve Wilson, who participated in the seminar,
Frank Raymond, Jack Mac Laughlin and W. Holstztynski.
Massachusetts Institute of Technology
TABLE OF CONTENTS
Chapter i. Preliminaries
i.i Modules ...................................... i
1.2 Homology and Cohomology with Twisted
Coefficients ................................. 2
1.3 A-Sets ....................................... 4
1.4 Microbundles, Block Bundles and Spherical
Fibrations ................................... 7
1.5 The Immersion Classification Theorem ......... ll
1.6 Intersection Numbers ......................... 14
1.7 Algebraic K-Theory ........................... 19
1.8 Localization ................................. 23
Chapter 2. Whitehead Torsion 28
Chapter .3 Poincare Complexes 39
3.1 Poincare Duality ............................. 39
3.2 Spherical Fibrations and Normal Maps ......... 45
Chapter 4 Surgery with Coefficients 54
4.1 Surgery ...................................... 54
4.2 The Problem of Surgery with Coefficients ..... 57
4.3 Surgery Obstruction Groups ................... 60
4.4 The Simply Connected Case .................... 74
4.5 The Exact Sequence of Surgery ................ 80
Chapter 5. Relative Surgery 82
5.1 Handle Subtraction and Applications .......... 82
5.2 Geometric Definition of Surgery Groups ....... 85
5.3 Classifying Spaces for Surgery ............... 94
5.4 The Periodicity Theorem, Part I .............. 96
Chapter 6. Relations Between Surgery Theories I01
6.1 The Long Exact Sequence of Surgery with
Coefficients ................................ i01
6.2 The Rothenberg Sequence ..................... 105
6.3 The Periodicity Theorem, Part II ............ 109
6.4 Simple Linking Numbers ...................... Ii0
Appendix A. Torsion for n-ads 122
Appendix B. Higher L-Theories 124
Appendix C. L Groups of Free Abelian Groups 127
Appendix D. Ambient Surgery and Surgery Leaving a
Submanifold Fixed 129
Appendix E: Homotopy and Homology Spheres ................. 135
References ................................................. 138
Symbol Index ............................................... 154
Index ...................................................... 156
Chapter i. Preliminaries
i.i. Modules.
Let A be a ring (not necessarily commutative)
with involution, i.e. a map A ÷ A, written ~ k ~*, so
that
(a) (~i+12)* = ll* + ~2'
(b) (Ii~2)* = ~2'~i*
(c) ~** = ~.
We will usually assume E i A. A denotes the group of
units in A. Unless otherwise stated, all A-modules will
be finitely generated and right. Let M be a A-module.
Then M inherits a left A-module structure by defining
l.m=m.l*.
The dual of M is defined by M* = HomA(M,A)
with A-module structure given by (f'~)(m) = l*f(m),
f~ M*, l~ A. If M and N are A-modules, we define M ~N
by giving N a left A-module structure as above. In the
case N=A, M @AA is a A-module with (x @ l)W = x@ ~*l.
A A-module is free if it is isomorphic to a direct
sum of copies of A. M is projective if there is a A-module
N so that M @ N is free. M is stabl[ free (s-free) if
we may choose N to be free, i.e. M ~ k A is free for some
k. If M is s-free, a stable basis (s-basis) is a basis
for some M@A k.
The main example of A will be a group ring,
A = Rw for some (usually commutative) ring R with i,
a multiplicative group with a homomorphism w:w ~ {±l}.
The ring Rw is defined to be the set of all finite sums
~ng.g, ngg R, ge w. The involution is given by
(~ng.g)* = ~w(g)ngg -1
1.2. Homology and Cohomology with Twi.sted Coefficients.
Let X be a finite CW-complex, ~ = ~l(X) and w:~ ÷ {±i}
a homomorphism. Let A = H~ and M a A-module. Define
~{i(x;M) = HI(C,(X) )~.~,,@
I} i(X;M) = H i(HomA(c ,(X),M)),
where C,(X) is the chain complex of cellular chains in
the universal cover X; C,(X) is a chain complex of free
and based A-modules. If X is not compact, we use cohomology
with compact supports. We write Hi(X) , Hi(x) for
HI(X;A), Hi(X;A).
We can define this alternately as follows: w
determines an element in HI(x;~/2H) and so a double cover
E ÷ X. Let Z/2H act on H non-trlvially and define t H
to be the bundle associated to E with fiber H. Now
÷ X is a principal x-bundle and so define ~ to be the
associated bundle with fiber M. Let ~t = ~ ~ zt. Then
Hi(X;M) = Hi(X;~ t)
Hi(x;M) = Hi(x;~)
where we use bundle (or sheaf) homology and cohomology.
If c is an n-cell in X, cap product defines
en:cq(x) ÷ Cn_q(X) @ A ~ Cn_q(X)
where cq(x) ~ HomA(Cq(X),A). This extends linearly to
chains in Cn(X) and, in fact, to infinite chains since we
are using compact supports.
This defines ~N:Hq(X) + Hn_q(X) for ~E Hn(X).
If M is a A-module, define ~:Hq(X;M) + Hn_q(X;M) by
the composition
HomA(Cq(X),M) ~ cq(x) @^M ÷ Cn_q(X)@^M.
If f:X ÷ Y is a map, f#:Wl(X) ~ Wl(Y) , then define
Ki(X;M ) = ker(f,:Hi(X;M ) ÷ Hi(Y;M))
Ki(X;M) ~ coker (f*:Hi(y;M) ÷ Hi(X;M)).
The condition wI(X) ~ Wl(Y) isn't necessary, but will
suffice for our purposes.
A map f:X + Y with f#:Wl(X) ~ Wl(Y) is a
homolo6~equivalence over R if f,:H,(X;Rw) + H,(Y;Rw)
is an isomorphism. X and Y have the same R-homology
tFpe if there is a sequence X = Z0,ZI,...,Z m = Y and
homology equivalences over R~ i ÷ Z Zi+ I or Zi+ I ÷ Zi~ for
each i.