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North-Holland Mathematical Library Board of Advisory
Editors :
M . Artin, H . Bass, J. Eells, W . Feit, P. J. Freyd, F. W . Gehring, H . Halberstam,
L . V . Hôrmander,
M . Kac, J. H . B. Kemperman,
H . A . Lauwerier, W . A . J. Luxemburg, F. P. Peterson, I . M . Singer and A . C. Zaanen
V O L U M E 26
N O R T H - H O L L A N D PUBLISHING C O M P A N Y AMSTERDAM NEW YORK OXFORD
Shape Theory The Inverse System Approach S. M A R D E S I C University of Zagreb,
Zagreb
Yugoslavia
J. S E G A L University of Washington Seattle, U.S.A.
NH
ψ 1982 N O R T H - H O L L A N D PUBLISHING C O M P A N Y AMSTERDAM NEW Y O R K OXFORD
© N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y — 1982 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 86286 2
Published by:
North-Holland Publishing Company — Amsterdam · New York · Oxford
Sole distributors for the U.S.A. and Canada:
Elsevier Science Publishing Company, Inc. 52 Vanderbilt Avenue New York, N . Y . 10017
L i b r a r y of C o n g r e s s C a t a l o g i n g in Publication D a t a
MardeSic, S. ( S i b e ) , Shape t h e o r y .
1927-
( N o r t h - H o l l a n d mathematical l i b r a r y ; v . 26) B i b l i o g r a p h y : p. Includes index. 1. Shape t h e o r y ( T o p o l o g y ) I . S e g a l , Jack. . I I . Title. I I I . Series. QA6l2.7.S^3 51k .2k 81-16809 ISBN O - W - 8 6 2 8 6 - 2 AACR2 1
P R I N T E D IN T H E N E T H E R L A N D S
T o Vera and A r l e n e
PREFACE
Shape theory is a new and important branch of geometric topology. Its rapid development is due to the fact that it affords a natural way to deal with homotopy properties of general spaces, in particular, of metric continua.
Even before the advent of shape theory in 1968 various
shape-theoretic ideas appeared sporadically in several areas of geometric topology. Examples of such notions are cell-like sets and the various UV-properties. T h e main purpose of this book is to present a systematic introduction to shape theory providing necessary background material, motivation and examples. H o w e v e r , we assume some basic facts of general and algebraic topology. T o date the only book on the subject is that of K . Borsuk [22]. There are also several sets of notes ([Borsuk, 16], [Segal, 6 ] , [Dydak and Segal, 2], [Mardesic, 20], [Cordier and Porter, 1] and [Summerhill, 1]). Our book has little overlap with Borsuk's. One of the special features
is the
systematic use of the inverse system approach. This approach has the advantage of applying to the most general spaces and yet is convenient for computations. Special attention is paid to compact spaces because shape theory is most useful there. W e devote considerable care to inverse limits and the related notion of resolution, even beyond the needs of shape theory, because of our continuing belief in their importance. Resolutions are a new tool, which for the shape of non-compact spaces play the same role as inverse limits do in the compact case. W e have used A N R expansions and polyhedral expansions since both are widely used and have their own merits. In laying the foundations of shape theory, w e were guided by the desire to introduce the topic thoroughly and thus make it accessible. Therefore we devoted to it all of Chapter I , steering a middle course
between being too abstract (categorical) and
too
concrete
(geometrical). In Chapter I I , we deal primarily with the algebraic topology of shape theory (shape invariants). W e concentrate on a number of fundamental notions like homotopy pro-groups, shape groups, movability, stability, and fundamental
theorems
like the shape versions of the
classical
Hurewicz and Whitehead theorems. W e also treat in detail algebraic vii
viii
PREFACE
requisites like the pro-groups, the Mittag-Leffler property and the first derived limit lim (including the non-abelian case). 1
In order to offer a more complete picture of the subject and adhere to space limitations, we present in Chapter I I I a number of surveys of selected areas of shape theory. These are meant to orient the reader and also serve as a guide to the literature. Proofs are discussed but not given. W e wish to point out that in the surveys we treat some of the most important achievements of shape theory. T h e reason why these results are not treated in full in this book is that they depend on theories and techniques from other branches of topology and have already been adequately presented elsewhere. The purpose of Appendix 1 is to present the relevant properties of polyhedra so as to make the book as self-contained as possible. In particular, we give a proof of the fact that the class of spaces having the homotopy type of polyhedra and those having the homotopy type of A N R ' s coincide. W e also treat the Cech system of a topological space. In Appendix 2 we give the original Borsuk description of shape for compact metric spaces and show that it yields a theory equivalent to the one presented in the book. H o w e v e r , acquaintance with this material is not essential. Extensive bibliographic notes follow each section. Internal referencing is by chapter, section and subsection (e.g., I I , §3.1, shortened to §3.1 if one is already in Chapter I I ) . External referencing is by author and entry (e.g., [Borsuk, 22]). T h e bulk of the work on the book was done during 1979/80 while Jack Segal (on leave from the University of Washington) was visiting the University of Zagreb under an exchange program between the National Academy of Sciences and the Yugoslav Council of Academies. This support is gratefully acknowledged. W e also want to thank our dedicated typists Bozena Grdovic and Pat Monohon. Zagreb and Seattle
Sibe Mardesic
October 1, 1980
Jack Segal
INTRODUCTION
Shape theory is a new branch of topology. L i k e homotopy theory, shape theory is devoted to the study of global properties of topological spaces. H o w e v e r , the tools of homotopy theory are of such a nature that they yield interesting results only for spaces which behave well locally (e.g., CW-complexes and A N R ' s ) . O n the other hand the tools of shape theory are so designed that they also yield interesting results in the case of bad local behavior. Moreover, shape theory does not modify homotopy theory on CW-complexes and A N R ' s , i.e., it agrees with the latter on such spaces. O n e cannot ignore spaces with bad local properties since, for instance, they arise naturally as the fibers of maps between spaces with good local properties. W e shall now illustrate the inadequacy of homotopy theory for spaces with bad local behavior by two examples. T h e first one concerns the well-known Whitehead theorem: If a map / : (X, * )—> ( Y, * ) of connected CW-complexes
induces
f #:
η
n
isomorphisms
of
all
homotopy
groups
π ( X , * ) — » 7τ ( Y , * ) , then / is a homotopy equivalence. η
If we omit the restrictive condition that X and Y be CW-complexes, the statement is no longer true as shown by the following example. L e t X be the
Warsaw
circle
Σ ÇR
2
given by Fig. 1, where a — (1/n, 1), n
α » = ( 0 , 1 ) , b = (1/π,Ο),fcoo= ( 0 , 0 ) , c = (0, - 1), d = (1, - 1). L e t Y be a n
point * and let / be the only map Σ —>*. Since Σ fails to be locally connected at the points of segment [ a , b*], it is readily seen that every œ
map ( § , * ) - » ( £ , * ) is inessential, i.e., π (Σ, * ) = 0 = π (*, * ) . Conse n
η
η
quently, / « # is an isomorphism for all η = 0 , 1 , 2 , . . . . Nevertheless, / fails
c Fig. 1. xiii
xiv
INTRODUCTION
to be a homotopy equivalence since Σ is not contractible. T h e latter fact can be seen by observing that the Cech cohomology Η\Σ)
= Ζ ^0.
A s a second example we consider the homotopy lifting property ( H L P ) of a map ρ : Ε —> B : Whenever Λ : X - » Ε and Η : X x / - » Β are maps such that /?Λ = H/o, where i (x) = (x,0), 0
lifting H , i.e., a map H.XxI—>E
then the homotopy Η admits a such that the following diagram
commutes.
X X /
T h e Hurewicz fibrations, which play an important role in the homotopy theory of CW-complexes, are defined as maps ρ which have the H L P . However, if Β has only constant paths (for instance if Β is the pseudoarc), then H(x, t) = H(x, 0 ) for all t G / and therefore for such a Β every map ρ : Ε — » Β is a fibration. This shows the inadequacy of this notion for such spaces. Hoping to overcome these difficulties of homotopy theory K . Borsuk undertook in 1968 the task of developing shape theory. W e outline here a line of reasoning which motivates and leads to some basic ideas of shape theory. One expects shape theory to yield a classification of metric compacta, weaker than homotopy type, but coinciding with it when applied to A N R ' s . In particular, one expects the Warsaw circle Σ to be in the same shape class as S because of a certain global similarity. Although 1
there is no essential map / : S - > Σ, there are maps of S into arbitrarily 1
1
small annular neighborhoods V of Σ in R , which all wrap S once around 2
1
V. This suggests that one consider, instead of single maps / : S —> Σ, 1
families of maps φ
of S into neighborhoods V of Σ. Clearly, it suffices 1
ν
to consider any decreasing sequence of neighborhoods V
n
with Π V = n
Σ, because such a sequence is a basis of neighborhoods of Σ. M o r e generally, every metric compactum Y can be thought of as being embedded in the Hubert cube Q or some other A N R M . Then one can consider a decreasing sequence of neighborhoods V of Y in M with n
Π V = Y Instead of maps / : X — » Y one considers sequences (φ ) of n
η
maps φ : X — » V η
n
such that for m ^ H, W η
n
η
in V . n
Unfortunately,
which maps into neighbor
hoods of a compactum Z , we can not compose them since the composi tion φ φ η
η
is not defined. In attempting to overcome this difficulty one first
xv
INTRODUCTION
notices that the neighborhoods V can be assumed to be compact A N R ' s . n
Therefore, every map φ : X — » V extends to a map / „ of some compact η
ANR
neighborhood
Π U = X and that f n
m
n
U
of X
n
into
V „ . O n e can also assume that
\ U — f in V , m ^ n. N o w the sequences of maps n
n
n
( / „ ) for X and Y and ( g „ ) for Y and Ζ compose termwise to yield a sequence ( g „ / „ ) for X and Z . One observes that the U„'s and the inclusions U — » l / „ , m ^ n, form m
an inverse sequence of compact A N R ' s with inverse limit X . Likewise, the V „ ' s form such an inverse sequence with inverse limit Y This shows that with X and Y we have associated inverse (inclusion) sequences of compact A N R ' s (U ) and ( V „ ) with inverse limits X and Y respectively n
and that maps X —> Y have been replaced by diagrams [/, < — rj < 2
which commute up to homotopy. One should also note that two sequences ( / „ ) , (f' ) for X and Y may be n
considered equivalent provided every η admits an m ^ η such that / „ I U ^f' \U m
n
m
in
V. n
In Chapter I we shall describe these ideas with more precision and in considerable generality and thereby obtain the shape category of topolog ical spaces. This category has as objects topological spaces and as morphisms certain maps between inverse systems of A N R ' s associated with the spaces in a particular way. Isomorphic objects in the shape category will be said to have the same shape. Moreover, a functor from the homotopy category to the shape category will be defined. T h e existence of this functor,
called the shape functor,
shows that the
classification of spaces according to shape is weaker than that according to homotopy type. It will also be clear from the definitions that the shape category restricted
to A N R ' s or CW-complexes coincides with the
homotopy category restricted to these spaces. Some of these considerations are purely categorical and therefore will be described on an abstract level first. Shape as a modification of homotopy will be discussed in the remaining part of the book.
CHAPTER I
FOUNDATIONS OF SHAPE THEORY
CONTENTS OF CHAPTER I
§1.
1. 2. 3. 4. §2.
§3.
§4.
§5.
§6.
3
PRO-CATEGORIES
Inverse systems Systems with cofinite index sets Level morphisms of systems Generalized inverse systems
3 9 12 14
ABSTRACT SHAPE
18
1. 2. 3. 4.
18 22 25 30
Inverse system expansions Dense subcategories T h e shape category Shape morphisms as natural transformations
ABSOLUTE NEIGHBORHOOD
RETRACTS
33
1. A N R ' s for metric spaces 2. Homotopy properties of A N R ' s 3. Pairs of A N R ' s
33 39 42
TOPOLOGICAL SHAPE
45
1. Shape for the homotopy category of spaces 2. Some particular expansions 3. Shape of pairs. Pointed shape
45 49 51
INVERSE L I M I T S A N D S H A P E OF C O M P A C T A
54
1. 2. 3. 4.
54 58 65 68
Inverse limits in arbitrary categories Inverse limits of compact Hausdorff spaces Shape of compact Hausdorff spaces Compact pairs
RESOLUTIONS OF SPACES A N D S H A P E
73
1. 2. 3. 4. 5.
73 76 81 84 86
Resolutions of spaces Characterization of resolutions Resolutions and inverse limits Existence of polyhedral resolutions Resolutions of pairs
§1. PRO-CATEGORIES
1. Inverse systems A preordering on a set Λ is a binary relation ^ on A which is reflexive and transitive, i.e., λ ^ λ
for each λ G Λ ,
λ s£ λ ' and A ' ^ A "
implies Α ^ Α " .
A preordered set (Α, ^ ) is said to be directed provided for any \
A G A
u
2
there exists a λ G Λ such that Α,^Α
and
A ^A. 2
A preordering is called an ordering if it is antisymmetric, A ^ A ' and Λ ' ^ Λ
i.e., if
implies A = A ' .
A n ordering =^ is linear (or total) if for any two elements A i , A G A 2
either Ai ^ A or A ^ Ai. A linear ordering is always directed. 2
2
E X A M P L E 1. Ν with its standard ordering is a linearly ordered set. E X A M P L E 2. A l l neighborhoods U of a point JC in a topological space X 0
form a directed ordered set provided U ^ x
U means U\D 2
U. 2
E X A M P L E 3. A l l open coverings % of a topological space X directed set provided
form a
^ °U means that % refines °ii . This preordering 2
x
generally fails to be antisymmetric, i.e., to be an ordering. If (Α, ^ ) is a preordered set and A ' Ç A , then ^ induces a preordering on A ' and one speaks of the preordered subset (A
^ ) . T h e preordered
subset A ' is cofinal in A if each A G Λ admits a A ' G A ' such that A ^ A '. If A' is cofinal in Λ , then A is directed if and only if A' is directed. R E M A R K 1. If ( Λ , ^ ) is a preordered set, one can introduce an equivalence relation ~ by putting Αι ~ A provided Αι ^ A and A ^ Ai. Let A' QA 2
2
2
be
a subset which contains precisely one element from each of the equivalence 3
4
[ C H . I, §1.1
FOUNDATIONS OF SHAPE T H E O R Y
classes. It is readily seen that ( Λ ' , =^ ) is a preordered subset of ( Λ , ^ ) and the preordering is antisymmetric. Moreover, A' is cofinal in A. Let ^ be an arbitrary category. A n inverse system in the category consists of a directed set A, called the index set, of an object Χ from % for λ
each λ Ε Λ and of a morphism ρ Moreover, one requires that ρ λ'^λ"
implies ρ >ρ >χ» = ρ ». λλ
λ
λλ
λ λ
λ λ
: Χ X , n
m > η + 1, are just compositions p
nm
=p
nn+l
·-p -i . m
An
m
inverse system indexed by a singleton is called a rudimentary system and is denoted by ( X ) , where X is its only term. A
morphism
of inverse systems X = ( Χ , ρ λ A ) — > Υ = (Υ ,q^>, λ
λ
consists of a function φ : Μ—>yl and of morphisms /
M)
μ
μ
:Χ
φ ( μ )
—» Υ
μ
in
one for each μ Ε M , such that whenever μ ^ μ ', then there is a λ Ε Λ , λ ^ φ ( μ ) , φ(μ%
for which
i.e., the following diagram commutes. X ( Y ) between the rudimentary systems. It is readily seen that this procedure embeds % in p r o - ^ as a full subcategory, i.e., takes distinct objects (morphisms) into distinct objects (morphisms) and every morphism ( X ) - * ( Y ) in p r o - ^ is obtained from a morphism X — » Y in
2. Systems with cofinite index sets W e say that a preordered set ( Λ , ^ ) is cofinite (or closure finite) provided for each λ Ε. Λ the set of all predecessors of λ is a finite set. E X A M P L E 4. Let S be an arbitrary set and let Λ be the set of all finite subsets F of 5 ordered by inclusion, i.e., Fi ^ F provided Fj Ç F . Then 2
2
(Λ, ^ ) is cofinite and directed. Another example of a cofinite directed set is (N, ^ ). The advantage of cofinite subsets is that one can use induction on the cardinal of the set of predecessors. This is illustrated by the proof of the following useful lemma. L E M M A 1. Let ( M , ^ ) be a cofinite preordered set and let ( Λ , ^ ) be a directed set. Then for every function φ : Μ — > Λ there exists an increasing function φ : Μ—> Λ such that φ ^ φ. PROOF. W e define φ by induction on the number of predecessors of μ G M distinct from μ. If μ
has no predecessors^ μ
we put φ(μ)=
φ(μ).
Assume that φ has been defined for all μ Ε M with < k predecessors distinct from μ, k ^ 1, in such a manner that μ ^ μ ' implies φ(μ)^ and φ(μ)^
φ(μ).1ΐ
φ(μ')
μ E M has k predecessors μ , , . . . , μ* distinct from μ,
then each μ, has (μ«) is
already defined. W e now choose for φ(μ)
a value ^ φ(μ ),..., ]
φ(μ^,
φ(μ). L E M M A 2. Let (/ , φ) : Χ —» Υ be a morphism of systems. If ( M , ^ ) is a cofinite directed set, then there exists a morphism of systems (g^, φ) : X — » Y μ
10
[ C H . I, §1.2
FOUNDATIONS OF SHAPE T H E O R Y
such that φ:Μ^Α
increases and for μ ^ μ '
the following
diagram
commutes. Χφ(μ')
(1)
Furthermore, ( g , φ) ~ ( / , φ ) so that (g^, φ) represents the same morphism M
f.X-^Yin
μ
p r o - ^ as ( / , φ ) . μ
PROOF. If μ ^ μ', then there is a λ ^ φ ( μ ) , Φ ( μ ' ) such that /μΡ Υμ
(4)
and we readily verify ( 1 ) . Finally, ( g , φ) ~ ( / , φ ) because of ( 4 ) . M
μ
R E M A R K 8. For systems over cofinite directed sets one can
describe
morphisms in p r o - ^ using morphisms of systems as in Lemma 2. This is a convenient simplification. T H E O R E M 2. Every inverse system X in p r o - ^ indexed by a set A admits an isomorphic system Y indexed by a directed cofinite ordered set M. each term (bonding morphism) in Y is actually a term (bonding
Moreover, morphism)
of X and the cardinal c a r d ( M ) ^ card(A ) . PROOF. It suffices to consider the case where A (Χχ,ρ
χχ
,A).
in infinite. Let X =
By Remark 5 one can assume that A is antisymmetric. W e
associate with X a new inverse system Y = (Y^,q^,M)
as follows. T h e
elements of the set M are all finite subsets of A, which have a maximal element max μ (which must be unique). Clearly, c a r d ( M ) = c a r d ( A ) . W e
C H . I , §1.2]
PRO-CATEGORIES
11
order M by putting μ ^ μ whenever μ , ς μ . ( M , ^ ) is a directed ordered χ
2
2
set. Indeed, if μ , , μ ε Μ then we choose an element λ Ε Λ such that 2
max μ
max μ =^ λ. Clearly,
Η
2
μ = μι U μ U { λ } Ε Μ
and
2
μ ,μ ^μ. {
2
A l s o notice that ( M , ^ ) is cofinite because a finite subset of A has only finitely many subsets. For μ Ε M we put Υ
=
μ
put q^>
X
m
a
x
and for μ ^ μ ' we
M
f^max μ., max μ. (notice that μ ^ μ ' implies max μ ^ m a x ^ ' ) . Then
F = ( Υ , q^>, M ) is an inverse system over a cofinite directed ordered set. μ
We
now define a morphism / : X — » F .
φ ( μ ) = max μ and let /
μ
= l
:X
m a x M
m
a
x
Let φ : M —» Λ be given by
^ ^ Υμ. Then ( / , φ ) : X — » F is a μ
morphism of systems because μ ^ μ ' implies Pmax μ, ηΐΗχμ'
^μμ'/μ'
/μΡφ(μ )φ(μ')
·
W e also define a morphism g : F — » X given by ( g , ψ ) . Let φ \ A
be given by ψ ( λ ) = { λ } .
Let g = 1 : Y A
A
{ A }
= X -»X . A
A
A—>M
If λ ^ λ ' , then
{ λ , λ ' } Ε M and max{A, λ ' } = λ '. Hence PA 'g '^{ '}.{ , '} A
A
A
A
=
A
Ρλλ' — g ^{ },{A, '}. A
A
A
Thus, ( g , ψ ) : Y ^ X is indeed a morphism of systems. A
N o w we consider the composites gf and fg. Since g / ^ A
Χ —>Χ , λ
λ
we see that
/μ&Μμ^ΐΓΠΗΧμΚμ
=
1 max
μ
=
gf = 1 1μ
· Υμ
>
(A)
= 1 : Χφψ
Since
φφ(μ)
Υμ ,
that ( / , )(g , ψ ) ~ ( 1
SO
A
= {max μ } ^ μ, μ
A
(Α)
=
we have μ
, 1M
),
l.C,
R E M A R K 9. O n e can also assign to any morphism h : Χ — » Χ ' in p r o - ^ a morphism Λ : F — » F ' between the corresponding systems over cofinite sets as follows. Let (h , χ) be a representative of Λ . Then we define κ : Μ ' — » M x
by κ ( μ ' ) = {^(max μ ' ) } E M , μ ' Ε Μ ' , as well as h,
. V"
'*ηι;ιχ μ' · ^^(max μ')
^
:Υ
κ ( μ Ί
^> Υ
μ
by Α: = μ
X' max μ ' ·
It is readily seen that we obtain in this manner a functor p r o - — » p r o - ^ which is naturally equivalent to the identity functor 1 : p r o - ^ — » p r o - i . e . , the following diagram commutes in p r o - ^ . h
X
>X' (5)
12
[CH.
FOUNDATIONS OF SHAPE T H E O R Y
I, §1.3
3. Level morphisms of systems Let X = (Χλ,Ρλλ , A ) and Y = ( Υ , q Λ
kX
, A) be two inverse systems over
the same directed set A. A morphism of systems ( / , φ) is a /eue/ morphism λ
of systems (also called a teue/ preserving or special morphism) φ =l
A
provided
and for A ^ A ' the following diagram commutes. ΧΛ
,N)
νν
N. W e now define φ' = 1 :
are inverse systems over
Ν and fl = / , j w
N
— (\μμ· ·
) A
: Χ : = Χ - * Υμ = Y C . λ
If ^ ^ ι/', then /ι/Ρι,ι/'
=
ίμ.Ρφ(μ)λΡλ\' = /μΡ