Localization of nilpotent groups and spaces (Amsterdam NH 1975)(ISBN 0720427169)

167 Pages·1974·4.16 MB·English

by  Peter John Hilton

LOCALIZATION OF NILPOTENT GROUPS AND SPACES
LOCALIZATION OF NILPOTENT GROUPS AND SPACES
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NORTH-HOLLAND
MATHEMATICS STUDIES 15
Notas de Matematica (55)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Localization of Nilpotent Groups
and Spaces
PETER HILTON
Battelle Seattle Research Center, Seattle, and Case Western Reserve University,
Cleveland
GUIDO MlSLlN
Eidgenossische Technische Hochschule. Zurich
JOE ROITBERG
Institute for Advanced Study, Princeton, and Hunter College, New York
1975
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM * OXFORD
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@ NORTH-HOLLAND PUBLISHING COMPANY, - AMSTERDAM - 1975
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Library of Congress Catalog Card Number:
ISBN North-Holland:
Series : 0 7204 2700 2
Volume: 0 7204 2716 9
ISBN American Elsevier: 0 444 10776 2
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NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
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Table of Contents
Introduction v11
Chapter I. Localization of Nilpotent Groups
Introduction 1
1. Localization theory of nilpotent groups 3
2. Properties of localization in N 19
3. Further properties of localization 23
4. Actions of a nilpotent group on an abelian group 34
5. Generalized Serre classes of groups 43
Chapter 11. Localization of Homotopy Types
Introduction 47
1. Localization of 1-connected CW-complexes 52
2. Nilpotent spaces 62
3. Localization of nilpotent complexes 72
4. Quasifinite nilpotent spaces 79
5. The main (pullback) theorem 82
6. Localizing H-spaces 90
7. Mixing of homotopy types 94
Chapter 111. Applications of Localization Theory
Introduction 101
1. Genus and H-spaces 104
2. Finite H-spaces, special results 122
3. Non-cancellation phenomena 133
Bibliography 14 7
Index 154
V
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Introduction
Since Sullivan first pointed out the availability and applicability
of localization methods in homotopy theory, there has been considerable work
done on further developments and refinements of the method and on the study
of new areas of application. In particular, it has become quite clear that
an appropriate category in which to apply the method, and indeed--as first
pointed out by Dror--in which to study the homotopy theory of topological
spaces in the spirit of J. H. C. Whitehead and J.-P. Serre, is the (pointed)
homotopy category NH of nilpotent CW-complexes. Here a pointed space X
is said to be nilpotent if its fundamental group is a nilpotent group and
operates nilpotently on the higher homotopy groups.
For a given family P ofrationalprimes, the concept of a P-local
space is based simply on the requirement that its homotopy groups be P-local.
Thus a localization theory for the category NH requires, or involves, a
localization theory of nilpotent groups and of nilpotent actions of nilpotent
groups on abelian groups. This latter theory could be obtained as a by-product
of the topological theory (this is, in fact, the approach of Bousfield-Kan)
but we have preferred to make a purely algebraic study of the group-theoretical
aspects of the localization method. Thus this monograph is devoted toanexposition
of the theory of localization of nilpotent groups and homotopy types.
Chapter I, then, consists of a study of the localization theory of
nilpotent groups and nilpotent actions. It turns out that localization
methods work particularly well in the category N of nilpotent groups, in
the sense that we can detect the localizing homomorphism e: G + Gp by
meansof effective properties of the homomorphism e, and that localization
does not destroy the fabric of a nilpotent group. For example, the nilpotency
.
class of Gp never exceeds that of G, and G embeds in HG Chapter I
P P
also contains some applications of localization methods in nilpotent group theory.
v11
Vlll Introduction
Chapter I1 takes up the question of localization in homotopy theory.
We first work in the (pointed) homotopy category of 1-connected CW-complexes,
H1
and then extend the theory to the larger category NH of nilpotent CW-complexes.
This extension is not only justified by the argument that we bring many more
spaces within the scope of the theory (for example, connected Lie groups are
certainly nilpotent spaces); it also turns out that even to prove fundamental
theorems about localization in H1, it is best to argue in the larger category
.
NH
One may represent the development of localization theory as
presented in this monograph--as distinct from an exposition of its applications
to problems in nilpotent group theory and homotopy theory--as follows; here
Ab is the category of abelian groups.
Thus we start from the (virtually elementary) localization theory
in the category Ab of abelian groups. The arrow from Ab to N
represents the generalization of localization theory from the category Ab
to the category N of nilpotent groups. The arrow from Ab to H represents
1
the application of the localization theory of abelian groups to that of
1-connected CW-complexes. The remaining two arrows of the diagram indicate
that the localization theory in NH is a blend of application of the
localization theory in N and generalization of the localization theory in
The diagram (L) which, as we say, representsschematically our approach
H1.
to the exposition of the localization theory of nilpotent homotopy types, is,
of course, highly non-commutative!
Introduction 1x
In Chapter 111, we describe some important applications of
localization methods in homotopy theory. Naturally, our choice of application
is very much colored by our particular interests. We have concentrated, first,
on the theory of connected H-spaces, and, second, on non-cancellation
phenomena in homotopy theory. Localization methods have proved to be very
powerful in the construction of new H-spaces and in the detection of obstructions
to H-structure. We give a fairly comprehensive introduction to the methods
used and obtain several results. Again, it has turned out that there is a
close connection between concepts based on localization methods and the
situation,already noted by the authors and others, of compact polyhedra
exhibiting either the phenomenon
X V A N Y V A , X+Y,
or the phenomenon
X x A = Y x A , X$Y;
we describe this connection in some detail.
Given a localization theory in some category C (and a reasonable
finiteness condition imposed on the objects under consideration, for reasons
of practicality), one can introduce the concept of the genus G(X) of an
object X of C. Thus we would say that X, Y in C belong to the same
genus, or that Y € G(X), if X is equivalent to Y for all primes p.
P P
It turns out that in the category Ab (confining attention to finitely-
generated abelian groups), objects of the same genus are necessarily isomorphic;
however, no such corresponding result holds in the categories N, H1, NH. (In
N we again confine attention to finitely-generated groups; in H1 and NH,
we confine attention to spaces with finitely-generated homotopy groups in each
dimension.) Thus localization theory naturally throws up questions of the
nature of generic invariants; we embark on a study of these questions in this