Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
542 David A. Edwards Harold M. Hastings
Cech and Steenro...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
542 David A. Edwards Harold M. Hastings
Cech and Steenrod Homotopy Theories with Applications to Geometric Topology
Springer-Verlag Berlin. Heidelberg 9New York 19?6
Authors David A. Edwards Department of Mathematical Sciences State University of New York at Binghamton Binghamton, N.Y. 13901/USA
Harold M. Hastings Department of Mathematics Hofstra University Hempstead, N.Y.11550/USA
Library ot Ceagress Catalegiag ia Pabllcatioa Data E d w a ~ s , David h
1946-
t e ~ and Steenrod hcmotopy theories w~th applications to gecm~tz~ic topology. (Lecture notes in mathematics ; 542) Bibliography: p. Includes index. i. Homotopy theory. 2. Geometry, Algebrale. 3. Algebra, Homological. 4. Algebraic topology. I. Hastings, Harold M., 1946joint author. II. Title. III. Series: Lectume notes in mathematics (Berlin) ; 542. 0~3.ia8 vol. 542 [QA612.7] 510'.8s [514'.24] 76-40180
AMS Subject Classifications 55399
(1970): 14F99, 14G13, 55B05, 55D99,
ISBN 3-540-07863-0 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07863-0
Springer-Verlag New York 9 Heidelberg 9 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 w 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. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
to Marilyn and Gretchen
CONTENTS
81.
INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . . . .
1
w
BACKGR~
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4
w
82.1
Pro - Categories
82.2
Some u s e f u l categories
82.3
Model categories
w
Simpliclal Closed model cateEorles
w
H o m o t o p y theories of p r o - s p a c e s
9
. . . . . . . . . . . . . . . . . . .
. . . . . . . . .
THE M O D E L S T R U C T U R E O N P R O - SPACES
; . . . . . . . . . . . .
41
. . . . . . . . . . . . .
48
.......
53
~ ......
. . . . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . . . . . . . . . . .
56
83.1
Introduction
w
The h o m o t o p y theo~7 of
w
The h o m o t o p y theory of p r o - C
83.4
S u s p e n s i o n a n d loop functors, c o f ~ r a t l o n
flbration
4
. . . . . . . . . . . . . . . . . . . . . .
sequences
g3.5
Stmplicial
model
w
Pairs
w
Geometric Models
w
InJ - s p a c e s
CJ
. . . . . . . . . . . . . . . . .
57
. . . . . . . . . . . . . . .
71
9 * 9 . . . . . . . . . .
. . . . . . .
structures
.
.
and
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
.
94 107
113
. . . . . . . . . . . . . . . . . . . . . .
115
. . . . . . . . . . . . . . . . . . . . . . . .
125
VI
w
w
T H E H O M O T O P Y INVERSE L I M I T A N D ITS A P P L I C A T I O N S TO HOMOLOGICAL ALGEBRA
. . . . . . . . . . . . . . . . . . . . . . .
129
w
Introduction
. . . . . . . . . . . . . . . . . . . . . . .
129
w
The h o m o t o p y inverse limit
w
E x~
w
The d e r i v e d functors of the i n v e r s e limit:
w
Results on d e r i v e d functors of the inverse limit
w
A l g e b r a i c d e s c r i p t i o n of
w
T o p o l o g i c a l d e s c r i p t i o n of
w
Strongly
w
The B o u s f i e l d - K a n s p e c t r a l s e q u e n c e
w
H o m o t o p y d i r e c t limits
on pro-Sw
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
Mlttag-Leffler
THE A L G E B R A I C T O P O L O G Y O F
lim s
background
- - 9
139 143
. . . . . . . . . . . . . .
145
. . . . . . . . . . . . .
152
. . . . . . . . . . . .
162
. . . . . . . . . . . .
166
lim s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
w
Introduction
. . . . . . . . . . . . . . . . . . . . . . .
w
H o ( t o w - C,)
w
R e m a r k s on c o m p l e t i o n s
w
Some b a s i c functors
w
W h i t e h e a d and Stability Theorems
w
Strong h o m o t o p y and h o m o l o g y theories
versus
134
.....
pro-groups
pro- C
130
tow - Ho(C,) . . . . . . . . . . . . . . .
169 172 172 172
. . . . . . . . . . . . . . . . . . .
181
. . . . . . . . . . . . . . . . . . . .
184
. . . . . . . . . . . . . . . . . . . . . . . . .
187 206
VJ~
w
213
PROPER HOMOTOPY THEORY
w
w
w
w
Introduction
w
P r o p e r h o m o t o p y and ends
w
P r o p e r h o m o t o p y theory of u - compact spaces
w
Whitehead
w
The Chapman Complement Theorem
9 9 9 9 . . . . . . . . . . . . . .
theorems
........
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GROUP ACTIONS ON INFINITE DIMENSIONAL MANIFOLDS w 7.1
Introduction
w
The theory of s - m a n i f o l d s
w
The S t a n d a r d A c t i o n s
w
P r o o f of T h e o r e m (7.3.4)
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
STEENROD HOMOTOPY THEORY
and Q - m a n i f o l d s
........
214 220 225 228
233 233 233 237
. . . . . . . . . . . . . . . . .
240
. . . . . . . . . . . . . . . . . . . .
245
Introduction
w
S t e e n r o d h o m o l o g y theories
w
The V i e t o r i s functor
w
Proofs of Theorems
w
S p e c t r a l sequences
w
Dual~ty
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
(8.2.19),
(8.2.20) and (8.2.21)
.....
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
SOME OPEN QUESTIONS
213
. . . . . . . . . . . . . . . . . . .
w
REF~ENCES INDEX
- type
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
245 246 254 268 269 276 279
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
294
w
Inverse systems of topological
INTRODUCTION
spaces occur in many contexts in topology.
Some
examples are:
a)
Geometric Topology. inverse systems
One can associate
{U},
{U \X},
over the open neighborhoods then one can associate
to
over compact subsets of
b)
Algebraic Topology.
of X
to any embedding
{(Y \X, U \X)}, X
in Y.
its end,
etc., where
If
X
The Cech construction
completion constructions
associates
C
where
C
varies
to a topological
A similar construction
also associate
to have an appropriate
inverse systems of spaces. C
varies
in
The Postnikov and inverse systems of
to complexes.
It is often important
any category
U
X.
algebraic geometry leads to ~tale homotopy theory.
complexes
the
is locally compact,
e(X) = {X \C},
space)an inverse system of CW-complexes.
pro -finite
X c-->Y
In [Gro -I]
another category
Grothendieck
pro - C
indexed by "filtering categories"
category and homotopy category of showed how to associate
to
whose objects are inverse systems in
and whose morphisms
are so defined as to make
cofinal systems isomorphic.
In [Q -i] Quillen introduced
the notion of a model
category as an axiomatization
of homotopy theory on
and
category is a category tions, fibrations
C
C,
weak equivalences
C.
topology of
in
SS.
together with three classes of morphisms
and weak equivalences
homotopy category of
Top
Ho(C),
and it is natural to consider
is obtained from
C
One has a canonical functor pro -Ho(C)
This point of view goes back to Christie
called cofibra-
which satisfy "the usual properties." by formally inverting
In [A- M] Artin and Mazur developed
pro - Ho(C).
A model
the
the algebraic
pro - C
> pro - Ho(C)
as the homotopy category of [Chr].
The
pro -C.
But Christie also realized that
for some purposes category.
pro -Ho(C)
It can be shown that
model category structure on pro - SS
pro -Ho(C)
pro - C.
is not the homotopy category of a
The second author
[Has -i] has shown that
admits a natural model category structure with homotopy category,
Ho(pro -SS), lences.
was too weak and one really wanted a stronger
obtained from
Ho(pro -SS)
pro - S S
had previously appeared
problem for topological
spaces
model category structure on
[Pot - 2].
and compare
in Porter's work on the stability
Grossman
[Gros - i] has studied a coarser
Towers -SS.
In the first part of these notes Ho(pro -C)
by formally inverting level weak equiva-
(w167
Ho(pro -C)
we develop the algebraic
with
pro -Ho(C).
We also give applica-
tions to the study of the derived functors of the inverse limit. of these notes (w167
contains applications
In w
w
trick," described
In w adjoint
theory of model categories
says that all inverse systems
[Q -i]
is reviewed in w
we show that for nice closed model categories
C,
extending
pro - C
holim:
Ho(pro - C)
> Ho(C)
theorems for
lim s.
The basic algebraic topology of with
various Whitehead
pro -Ho(C), theorems.
C,
CJ
(where
J
is a
inherit natural closed model struc-
[Has -i].
we show that the natural inclusion
obtain vanishing
Ho(pro -C)
in w
to inverse systems indexed over cofinite strongly directed sets.
cofinite strongly directed set) and tures from
We conclude with a brief list of
contains background material about p r o - categories and model
The "Mardesic
are pro - equivalent Quillen's
group
in w
More precisely, categories.
The second part
to proper homotopy theory,
actions on the Hilbert cube, and shape theory. open questions
topology of
Ho(C)
~ Ho(pro - C)
(compare Bousfield and Kan
Ho(pro -C)
is developed
in w
has an [B -K]) and
We compare
discuss homotopy and homology p r o - groups, and prove w
Whitehead and stability theorems,
survey of work of the first author and R. Geoghegan
[E - G -i -5].
includes a
In w
we show that the category of o -compact spaces and proper maps may be
embedded in a suitable category of towers which is a closed model category.
We
then apply pro -homotopy theory to proper homotopy theory and weak -proper -homotopy theory (see [Chap- i] and [C -S] for weak -proper -homotopy theory and its uses in the study of Q -manifolds and shape theory.)
Some of our results are announced in
[m -H - 3 ] . We apply this theory in w manifolds.
In w
w
to the study of group actions on infinite dimensional
represents joint work with Jim West [West-l],
[ E - H -W].
we discuss strong shape theory and develop generalized Steenrod homology
theories using pro -homology.
These theories have found recent applications in the
Brown-Douglas-Fillmore theory of operator algebras Kaminker and Schochet [K -S],
and
[B - D - F -i -2]; see also
with D. S. Kahn,
[K - K - S].
Detailed introductions precede w167
Acknowledgements.
We wish to acknowledge helpful discussions with Tom Chapman,
Ross Geoghegan, and Jim West, and correspondence with A. K. (Peter) Bousfield and Jerry Grossman.
Some of this material was presented at conferences at Syracuse
University (Syracuse, N. Y., U,S.A., December, 1974 and April, 1975), Mobile, Alabama, U.S.A., (March, 1975), New York (March, 1975), the University of Georgia, U.S.A., (August, 1975), Guilford College (Greensboro, N. C., U.S.A., October, 1975), and the Interuniversity Center in Dubrovnik, Yugoslavia (January, 1976).
We wish
to acknowledge the organizers of these meetings for their hospitality.
The second-named author held a visiting position at the State University of New York at Binghamton during the academic year 1974-75 in which much of this work was done.
He wishes to acknowledge their support and hospitality.
He was also
partially supported by N.S.F. Institutional Grants at Hofstra University in 1973-74 and 1975.
We wish to thank Althea Benjamin for typing this manuscript.
w
w
BACKGROUND
Pro -Categories. We need a Category of inverse systems such that cofinal subsystems are isomor-
phic; such a category was first defined by Grothendieck in [Gro -1] and is described in detail in the appendix of [A-M].
(2.1.1)
Definitions.
A non-empty category
J
is said to be left filtering if
the following holds. a)
Every pair of objects
j,j'
in J
can be embedded in a diagram j l
j"
j
b)
If
j'--~> J
is a pair of maps in
such that the compositions
If
C
X:J --~C.
and J
are categories, then a
>j
then there is a map
pro -object
filtering category.
~-dia~ram
over
C
in J
is just a functor
suppressing both
J
and the
X(j) X(F)> X(j').
over The
C
is a
J -diagram over
pro -objects over
C
C
where
J
form a category
defined by (pro - C) ({Xj }, {Yk }) - llmkColim j CC(Xj ,Yk ) }.
(Note:
j"--->j'
are equal.
We will usually use the notation {Xj},
bonding morphisms
A
j"
J,
the indexing categories are not assumed equal.)
is a small left pro - C
with maps
Let
tow - C
be the full subcategory of
indexed by the natural numbers.
pro - C
Objects of
We have defined the set of maps in
consisting of those objects
tow - C
pro - C
are called towers.
from
X
to Y,
but the above
definition is somewhat opaque and it is not obvious how to define composition of maps from the above definition.
Hence, we shall give an alternate definition.
For simplicity, consider inverse systems directed sets
(2.1.2) map
J
some
j
> Yk in J
and
{Yk }
over
C
indexed by
respectively.
Definition.
8:K--~J
fk:Xs(k)
and K
{X.} 3
A morphism
f:X--~Y
in
pro - C
is represented by a
(not necessarily order -preserving) and morphlsms in C with
for each j ~ 8(k)
k and
in K
such that if
j ~ 8(k')
bond
k ~ k'
in K,
the diagram commutes.
then for Two pairs
fk '
Xj
> Xo(k')
~ Yk'
\
bond \ ~
bond fk
X~(k) (8,{fk})
and
k
there is a
in K
(8',{f~})
composite maps
(2.1.3)
factor through
represent the same morphism in
j
in J
fk o bond,
Remarks. pro - C.
with
j ~ 8(k)
fk' o bond : X.3 -'~Yk
and
A pro- object
{X]}
limj{Xj}
pro - C
j ~ 8'(k)
if for each such that the two
are equal.
The inverse limit functors
information than its inverse limit exist.
~"Yk
lim:C J --~C, in pro - C
in C;
The relationship between the pro - object
{X.} 3
if they exist,
contains much more
the latter need not even and its inverse limit
llmj{xj} point
is analogous
p
to the relationship
and its value at
between the germ of a function at a
p.
We shall need the following reindexing results from Artin and Mazur
(2.1.4)
Proposition.
isomorphism
A map
(in Maps (pro -C))
f. -J{X~ --/-~Yj},_
Maps (C).
the following holds.
Proposition.
and suppose that
A
Let
A
be a finite diagram with commutation relations,
has no loops, i.e., that the beginning and end of a chain of
arrows are always distinct.
Let
D
be a diagram in
i.e., a morphism of
A
to pro - C.
{Dj}
C
such that the diagram in
of diagrams of
isomorphic
to
can be represented up to
by a small left filtering inverse system of maps
i.e., by a pro -object over
More generally,
(2.1.5)
f:X---~Y E pro -C
[A -M].
pro - C
of the type of
There is a left filtering inverse system pro- C
determined by
{Dj}
is
D.
Our techniques often require that the indexing categories be cofinite element has finitely many predecessors) implies
A,
a = b).
strongly directed sets
The following relndexing
(a ~ b
(each and
b ~ a
trick was inspired by Mmrdesic
[Mar - i].
Let An object d,
I
be a small loft filtering category and d E D
and for each
objects,
D--~ I
will be called an initial object if d'
in D,
there is a unique map
if they exist, are clearly unique.
diagrams with initial objects over
I.
Let
D
a diagram over
contains no maps into
d --# d' M(1)
We shall call
I.
in D.
Initial
be the set of finite D ! D'
in M(1)
if
D
is a subdiagram of M(1)
D'.
M(1)
Further,
is a directed set. D
the initial objects of
is clearly cofinite.
and
D ~ D'
D'
and
are equal).
Because
D' ~ D
I
is filtering,
implies
Hence,
M(1)
D = D'
(and
is a cofinite
strongly directed set.
Define a rune,or D
in M(1)
D'
its initial object in
from the initial object of
required functor.
as follows.
inlt : M(1) --~ I
Further,
I.
D'
If
Associate
D < D'
to a diagram
there is a unique map in
to the initial object of
D.
This yields
the
the functor inlt is clearly coflnal.
We thus obtain a functor
M:pro - C
9 pro - C ,
with
M{Xi}i E I
= {Xlnlt(D)}D~ M(1)'
and a natural equivalence
init:X
Summarizing,
(2.1.6) equivalent
we have the following theorem.
Theorem.
There exists a funetor
to the identity,
directed set for every
(2.1.7) isomorphic j
in
X
such that
M(X)
An object
pro - C
to an object of
k > j
M:pro - C
~ pro - C,
naturally
is indexed by a eofinite strongly
in pro -C.
Definitions.
there exists a
the diagram
~ M(X).
{5} C.
such that for each
of pro - C {X.} 3 s > k
is called stable if it is
is called moveable
if for each
there exists a filler in
....
->Xs
\,/ X. J
The above description of of direct systems over inj -C
C
pro -C
in which cofinal systems are isomorphic.
inj -C
Morphisms in
are given by the formula
inj - C ({Xj },
As with {Xj}
may be dualized to yield a category
to
pro -C,
{Yk }
J = {j}
and
In this case a map may be represented by a function
fj:Xj
there exists a
we shall give an alternative description of a morphism from
in the case that the indexing categories
are directed sets. and maps
{Yk }) = limj colim k {C(Xj,Yk ) }.
> YO(j) k
with
in C
for each
k > 8(j)
and
j
in J
k > 8(j')
f~
such that if
K = {k} 8:J--~K
j < j'
such that the diagram commutes.
bond
Xj
~ Y8 (J)
" Yk
bond]
/~oL~d f1'
Xj. ~
Ye ( j )
All of the above theory of Mardesic construction, colim: CJ--->C,
(2.1.8) HOM: C ~
x C
pro - C,
including Artin-Mazur reindexing and the
may be dualized to
inj - C.
if they exist, factor through
Proposition. ~ C.
Then
Suppose that HOM
C
The colimit functors,
inj- C.
admits an internal mapping functor
extends to a functor
HOM: (pro- C)~
x C
,inj-C.
Proof.
The required functor is given by
HOM ({Xj},Y) = {HOM (Xj,Y)}.
w
Some useful categories. This expository section is intended as a reference for later sections.
We sug-
gest that the reader omit the proofs the first time through, and refer to this section as needed later.
We shall give brief sketches of the following categories used in the remainder of these notes.
SS,
the category of simplicial sets introduced by D, M. Kan [Kan -2] and
J. C. Moore,
See [May-l]
and [Q-I,
w
D, Quillen's closed model category
[Q -i], a category with sufficient structure to "do homotopy theory," is an abstraction of
CW
SS.
prespeetra and
CW
spectra
(CWSp)
introduced by J. Boardman.
We follow
J. F. Adams [Adams -i] for the "additive structure" and [Has -3 -5] for smash products.
Sp,
the category of simplicial spectra developed by D, M. Kan [Kan-l].
K, S. Brown [Brown] sketched a proof that
Sp
is a closed model category.
We introduce both categories of spectra because homology theories (which involve smash products) are most easily described in
CW
spectra, while the closed model
structure of simplicial spectra is needed to define a suitable homotopy category of pro -spectra.
M. Tierney [Tier] described stable realization and singular functors inducing an equivalence of homotopy categories
Ho(Sp)
~ Ho(CWSp).
D, W. Anderson [An-l,2]
gave a different construction of homology theories using chain functors.
J.P.
May
[May -2] introduced a category of spectra based on the Boardman-Vogt theory of infinite loop spaces [B -V] and thus described the higher structure of ring spectra.
10
While these approaches are more powerful, the "classical" approach we shall follow is simpler and adequate for our purposes.
We shall then describe how the
use
of spectra yields a unified treatment of
generalized homology and cohomology theories, Spanier-Whltehead duality, and homology and cohomology operations, following [Adams - i, 3].
We begin by defining simplicial sets.
(2.2.1)
a)
Definition.
A sequence {X,
A simpliCial Set
n ~0}
X
consists of:
The elements of
of sets.
X
are called n
the ~ - s i m p l i c i e s
b)
Face maps
c)
Degeneracy Maps
of
X.
d~:X--->X_I
for
n ~ i
and
for
n > 0
n
si:Xn--~Xn+l
0 < i < n.
and
0 < i < n.
The maps are required to satisfy the following identities:
did j -~ dj_ I d i
for
i < j,
si s.3 = Sj+l s.m
for
i < J,
sj_ 1 d i
for
i < m,
id
for
i = j
for
i 9 J+l.
~ dis.3 ~ ~
\
sj di_ 1
For example, the singular complex
S(X)
or
j+l,
of a topological space
simplicial set with typical n - simplex a continuous map the standard n - simplex in
Rn+l.
f:An--->X,
X where
is a An
is
11
(2.2.2)
Definition.
sequence of maps
fn-i di = difn
sets.
f:X--+Y n ~ 0,
f :X --->Yn' n n and
Definitions
A map
sets consists of a
which satisfy the identities
Sifn = fn+l s . . 3
(2.2.1) and (2.2.2)
SS
combine to yield the category
SS.
d0x,
Let
x
faces of
x,
example,
dj (djx) = dj (dj+ix) .
extension condition
j = 0,1,''',1,''',n.
A simpliclal
y0,Yl,''',yi,-'',yn,
with
fibration
djx = yj
More generally, b
a map
in B,
n
which satisfy the appropriate
for
for
with in SS
compatibility
and
of spaces induces a Kan fibration
[Kan -2] gave a combinatorial
n
description
[Mil -2] geometric realization
(=
~i(RX)),
functor.
where
(n- i) -
d.x = y. 3 3
for
is called a Kan in E,
conditions and
there is an n - simplex p(x) = b.
also a Hurewiez
S(p):S(E) ---> S(B).
of the homotopy groups of
This definition of the homotopy groups is extended ~i (x) ~ ~i (SRX)
for
to be
Kan complexes and also described a functorial Postnikov decomposition
by defining
The
conditions
(n - i) - simplices
j = 0,1,''',1,''',n,
j = 0,1,"',1,--',n,
X.
conditions,
if given any
x
p:E--->B and
set
of the
is said to satisfy the Kan
the singular complex of a space is a Kan complex;
p:E--->B
D. M. Kan
X
there exists an n - simplex
p(yj) = d3.b
For example,
set
which satisfy the compatibility
fihration if given an n - simplex
in E
satisfy certain compatibility
(or is simply called a Ka____nncomplex)
the faces of an n - simplex,
the requirement
be an n - simplex of a simplicial
dlX,-'',dnX ,
^ ... y0,Yl,-'',yi, 'Yn
simplices
plexes.
of simplicial
The following Kan extension condition is crucial to the development
homotopy theory of
x
of simplicial
of Kan com-
to all simpllcial
R:SS --->Top
sets
denotes Milnor's
12
Call a map (i.e., each
f:X-->Y
fn
of simplieial
is an inclusion),
a weak. equivalence if
and for every choice of basepoints homotopy-extension
sets a cofibration
in
X,
and covering-homotopy
~,(f)
if it is an inclusion
T0(f )
is a bijectlon,
is an isomorphism.
properties
The usual
are combined in the following
theorem.
(2.2.3)
Coverin~ Homotopy Extension Theorem.
Given a commutative
solid-arrow
diagram
9-
A
i>y ~
/
X
in which
i
is a coflbration,
equivalence,
is a flbration,
then there exists a filler
See, e.g. M1
p
>B
[Q-I,
w
[Q-I];
see w
i
or p
is a weak
f.
for a proof.
for a model category
and either
Theorem
(2.2.3) becomes Quillen's Axiom
.
Note that the usual homotopy extension property only holds for maps into Kan complexes.
(2.2.4)
Definitions.
finitely many non-degenerate has no non-degenerate
In w
A simplicial simplices;
simplices
set X
X
is called finite if
is said to have dimension
in dimensions
greater than
we shall discuss function spaces in
SS.
n.
X
has only ~ n
if
X
13
We shall now sketch the basic properties We shall follow J. F. Adams
of
CW
prespectra and
for the "additive structure"
[Has -3 -5] for smash products and function spectra.
evident basepoint.
namely
Let
spectra.
[Adams -i], except for a technical modification made in
[Has -3] to construct smash products,
ture with two vertices,
CW
0
and i.
Let
Give
and follow
[0,i]
the
CW
S 1 = [0,i]/0 ~ i,
S n = S 1 ^ S I ^ "'" ^ S 1
struc-
with the
(n factors), where
K A L = K x L/K v L.
(2.2.5) pointed
CW
Definitions.
A
complexes
X n ^ S 4 -->Xn+ 1
CW
spectrum
{Xnln = 0,1,2,...},
for each
n.
are
^ id CW
S 4"
The category
of
spectra and whose morphisms
A weak spectrum spaces,
CWPs
f :X nn
X = {X } n
CW
~Y
fn : X --->Yn
such that
The category cofinal inclusions
(2.2.6)
f:X-->Y
CWSp
fn+l
of
CW
such that
fn+l
extends
is the category whose objects
are prespectrum maps.
is a sequence of pointed compactly generated Xn ^ S 4---> Xn+ I
for each
n.
consists of a sequence of continuous pointed maps extends
fn ^ ids4
up to homotopy.
spectra is obtained from
[Adams -i].
called cofinal if for each cell
CWPs/{cofinal
consists of a
CWPs
by inverting
of spectra.
Definitions
Io u {basepoint}~
n
f:X--->Y
prespectra
together with continuous pointed maps
weak prespectrum map
consists of a sequence of
together with cellular inclusions
A prespectrum map
sequence of continuous pointed maps
fn
X = {X } n
A S 4k
inclusions}.
X' n+k"
o c X
A subspectrum
n
c X
X'
of a
there exists a
The category of
CW
CW k
spectra,
spectrum
such that CWSp,
is
X
is
14
(2.2.7) a)
Remarks.
The class of cofinal
inclusions
admits a calculus of right-fractlons
in the sense of P. Gabriel and M. Zisman any map
f:X--->Y
of
CW
X'
is cofinal in
such diagrams same map if
X ~ X'
X
and
f' ~ Y
f' = f"
on
This means that
spectra can be represented
X D X'
where
[G -Z].
by a diagram
f' ~ Y
f'
is a prespectrum map.
Two
ft!
and
X m X"
X' N X".
"~Y
represent
Composition
the
is defined as follows.
Consider a solid-arrow d i a g r a m
/
/
X I! \
/I
\ \f!, a Y'
X'
X
Y
Define a subspectrum of X', maps
X'
which
f'
X" c X'
maps into
hence cofinal in X D X'
f' > Y
X. and
This yields a w e l l - d e f l n e d
b)
Adams uses
S!
\
Z
as follows: Y'.
f" = f'IX". K' > Z
y D y'
composition
S4
c)
X"
is cofinal
is given by
and m a k e s
CWSp
of
yield equivalent
X 9 X"
K ' f " ~ Z.
a category.
CW
spectrum.
categories
of
spectra.
By applying realization cylinder
and cylinder
functor to a weak spectrum
functors X
in
The composite of the
in the definition
It is easy to see that both definitions CW
consists of those cells
By construction,
Let
w h e r e we use
X"
and a suitable m a p p i n g
w e obtain a
CW
spectrum
X'
15
X' --->X
and a natural p r e s p e c t r u m m a p
which is a weak homotopy
equivalence on each level.
(2.2.7)
a)
Some examples of
CW
The k -sphere spectrum
s~ -
fs~t
spectra.
(k ~ Z)
is given by
{*'
-
Sk + 4n(space ) ,
I
k+4n
>
I
together with the inclusions Sk + 4 n ( s p a c e ) ^ S4 ( s p a c e )
(sl^
.^s I)^CS I ^
^s I)
S 1 ^ ... ^ S 1
i S k + 4n .
b)
More generally, let
K
be a pointed
CW
complex.
Associate to
K
the spectrum
K
(spectrum) = K
ffi {K A S 4n}
,
together with the evident inclusions. K ^ Sk
for any
k E Z
(spectrum)
(space) ^ S 0
More generally, we may form
(see (2.2.8)).
suspension spectra; for sufficiently large (K A Skln ^ S 4 ~ (K ^ Sk)n+l. a stable finite
CW
complex.
If
K
These spectra are called n,
is finite, we call
K A Sk
E. Spanier and J. H. C. Whitehead
IS -W] first studied stable finite complexes.
18
c)
The Eilenberg-MacLane spectra.
Let
associated with the weak spectrum K(G,4n) ^ S4
~ K(G,4n +4)
cohomology class.
Here
G
K(G)
he the
{K(G,4n)},
maps
d)
The B_~U-spectrum: BU n ^ S 4 BU ^ S 2
~ BUn+ I ~ BU.
equivalences~ BO,
BU
BSO,
n
in X)
= BU
Xn
BSp,
A
The Eilenberg-
CW
> fl4Xn+I
spectrum
X
(adjoint to the
for all
n,
and the maps
are defined by composing Bott periodicity maps
is an ~ -spectrum. and
BSpin
"infinite Lie groups."
BU
BU
> ~2BU
are homotopy
There are similar spectra
associated with the respective classifies complex K -theory,
BO
real K - theory, etc., see (f) below.
e)
The Thom spectra classifying bundles e.g. [Sto]) DU(m)
in
Note that
MU(m) EU(m)
MU,
MO,
MSO,
MSp,
EU(m) ---~BU(m).
M spin.
Consider the
The Thom complex (see,
is the quotient of the unit disk bundle by the unit sphere bundle
MU(1) m D2/S 1 m S 2.
MU(s
is
are homotopy equivalences.
Because the adJoints BU
together with maps
is any abelian group.
if the maps
X n ^ S 4 ----->Xn+ I
spectrum
which classify the fundamental reduced
MacLane spectra are examples of ~ - ~ e c t r a . called a n ~ - s p e c t r u m
CW
^ MU(m) = DU(s
roT(9.
in EU(m).
There are Whitney sum. maps
• DU(m)/Sphl U(s
---> DU(% +m)/sph U(s +m)
=
Sph U(m)
+m).
X
Sph U(m)
17
These yield inclusions
MU(m) ^ S 2
and thus the
MU
spectrum
MU(m+ i),
MU = {MU(2n)}.
yield the other Thom spectra.
MU
Similar constructions
classifies complex cobordism by
a classical result of Thom, see e.g.,
[Sto].
Similarly, the other
Thom spectra classify appropriate cobordism theories. is important because:
Cobordism
(I) some varieties are quite powerful; and
(2) the dual homology theories, bordlsm, have natural geometric definitions.
f)
R. Stong [Sto] is a good source on cobordism.
The examples (c), (d), and (e) above classify well-known cohomology theories:
H*(;G),
various forms of K - theory, and various forms Brown's theorem associates to
of cobordism theory respectively.
E*
any generalized cohomology theory
a spectrum
E = {g } n
with
E4n(x) e [X, En].
(2.2.8)
Definition.
If
K
trum, then their smash product is K ^ Xn ^ S4
~ K ^ Xn+ 1
(2.2.9)
Definition.
there is a map
is a pointed
complex and
K ^ X = {K A Xn},
induced from
Maps
CW
X
is a
CW
spec-
together with inclusions
X.
f,g:X .[ ~Y
H:([0,1] u *) ^ X - - ~ Y
of
CW
such that
spectra are homotopic if HI(0 u ,) ^ X = f
and
H I(l U *) ^ X = g.
Let Ho (CWSp)
Ho(CWSp)
denote the resulting homotop7 category of
CW
spectra.
is essentially the category introduced by Boardman (see [Vogt -2]) as
formulated by Adams [Adams -I].
Caution:
CWSp
is no__~ta model category.
18
The homotopy ~roups of a for
CW
~k x = Ho(CWSp)(sk,x)
spectrum are given by
keZ.
(2.2.10)
Remarks.
full subcategory of
The homotopy category of stable finite Ho(CWSp),
CW
complexes, a
is the classical Spanler-Whitehead category (use
the Freudenthal suspension theorem which states that [X ^ Sn, Y ^ Sn] ~-[X A Sn+l, y A Sn+l] sufficiently large
and YM
X
finite if for some
N,
and Y
and
n).
More generally call a complex and
for finite complexes
Xn = ~
CW
spectrum
A S4 n - 4 N
for
X
n ~N.
If
X
and Y
~
is a finite
are finite, and
are the above complexes, the~
(2.2.11a)
CWSp(X,Y) -= c O I ~ k { C W [ ~
^ Sk - 4 N , YM A sk-4M~},
and
Ho(CWSp)(X,Y) a colimk{[~ ^ Sk - 4 N ,
(2.2.11b)
The category
CWSp
subspectra.
In fact, a
finite subspectra. morphisms of
CW
YM ^ Sk-4M]}"
is then defined so that a spectrum is the colimit of its finite CW
spectrum is the homotopy colimit (see w
J. Boardman (see [Vogt -2]) and A. Heller [Hel -1,2] define spectra by this criterion and (2.2.11a).
Cofibrations and homotopy equivalences behave similarly in complexes.
of its
CW
spectra and
A. Heller formalized these properties by introducing abstract
CW
h-c
categories [Hel -1,2].
We shall need smash products and function spectra in order to discuss generalized homology and cohomology theories.
These constructions are more difficult than the
19
above structure. unit [Has- 5].
In fact, there is no smash product on
Ho(CWSp)
spectra.
SO
as
J. Boardman (see [Vogt 2]) gave the first construction of a
coherently homotopy associative, commutative, and unitary CW
with
D.M.
(S0)
smash product for
Kan and G. W. Whitehead [K -W] described a non-associatiVe
smash product for simpliclal spectra.
Later Adams [Adams- I] and still later the
second author [Has - 3] gave different and simpler constructions for the smash product of
CW
spectra.
J.P.
May [May - 2] and D. Puppe [Puppe] gave a radically
different construction following the Boardman-Vogt theory of infinite loop spaces [B- V].
We shall follow [Has - 3].
(2.2.12)
Th_~einterchanse problem.
a (non-commutative) ring and any
CW
We may regard the sphere spectrum spectrum as a right-module over
Construction of a smash product requires permutations Because
S0
SO
as
SO = {s4n}.
~ of S 4 ^ .,- ^ S 4.
is only homotopy commutative, this requires canonical homotopies
H W
from
~
to the identity.
These are defined as follows.
S 4k _= S 4 A - - -
A S4 & C2.
A "'"
(C 2 x
where
denotes the one-polnt compactificatlon.
factors of
C 2 x 9 -. x C2;
thus
w ~ SU(2k)
"'"
Identify
A C2.
x C2) *
Then
w
Because
simply permutes SU(2k)
netted and simply connected, there is a unique homotopy class of paths SU(2k)
with
r(0)
= w
and
r(1)
H :([0,i] u *)
= e,
A
the identity of
S 4k ! [0,i] x S 4k
W
by K=(t,x)
= r (t)(x).
SU(2k).
- S 4k
is path con-
[r ] Define
in
20 Then
[H]
is the required homotopy class (relative to the endpoints)
of canonical
homotopies.
We shall now define a family of smash products on
CWSp,
and prove that they
are all equivalent and have the required properties.
(2.2.13)
Definition.
Given a sequence of pairs of non-negative integers
{ (in,J n) In => 0, and
{in }
in + i n ffi n,
and
{i n )
are monotone unbounded sequences},
define an associated smash product by
X^Y={(X^Yln}~{Xi
^ Yjn }
n
together with the appropriate inclusions induced from
Then
^
extends to bifunetors on
CWPs,
inclusions is a cofinal inclusion), and
(2.2.14)
Theorem.
smash products on
CWSp
X
and Y.
(the smash product of cofinal
Ho(CWSp).
Any two sequences (2.2.13) yield canonically equivalent
Ho(CWSp).
We shall need the following machinery.
(2.2.15)
Definition.
Let
X
be a
CW
spectrum.
Given a monotone unbounded
sequence of non-negative integers
{Jn[n >__O, define a
CN
Jn < n),
spectrum 4n- 4Jn } DX = ((DX)n} = {Xjn ^ S
together with the appropriate inclusions induced from
X.
21
Then
D
extends to a functors
natural cofinal inclusions
(called destahillzation)
DX--->X,
so that
D
on
CWSp.
There are
is naturally equivalent
to the
identity.
(2.2.16) of DX
Definition.
Let
X
be a
CW
prespectrum.
A permutation
consists of a sequence of maps
{~n = id a ~n:
(DX)n
=
XJn
-'+X.
^ S4 ^ 9
^ S4
A S4 A 9
^ S4
3n
(DX)n} where each
n
S 4 ^ ... ^ S 4.
is a permutation of
A sequence of maps
g = {gn: (DX)n ----~Yn]gn+l
extends
gn
up to permutation}
is called a permutation map.
(2.2.17) homotopies
Proposition.
Permutation maps are weak maps, where the required
Hn:(DX)n A S 4 x [0,i] ---->Yn+l
are induced from canonical homotopies
(2.2.12).
The proof is easy and omitted.
The following lemmas relating weak maps and maps of [Has -3].
The proofs involve construction
CW
spectra are proved in
of suitable mapping cylinders and the
homotopy extension property.
(2.2.18) for
Lemma.
A weak map
f:X-->Y,
together with a family of homotopies
f,
{H:X
^ S 4 • [0,1J
~Yn+l}
22
(see (2.2.5)), induces a strict map
F
depends upon
{f}
and
F:S--->Y.
{H }
up to equivalence in
Ho(CWSp).
n
(2.2.19)
Lemma.
Let
{H } and
{H'}
n
map
f.
If
and
(f,(Hn})
(2.2.20) topies
{H'} n
Hn = H'n
relative to the endpoints for each
induce homotopic maps
Lemma. and
be families of homotopies for a weak
n
Let {H" } n
X--->Y
f':X--->Y
and
respect•
in
n,
then
(f,{Hn})
CWSp.
f":Y--~Z
be weak maps with homo-
Define "composed" homotoples
Hn:Xn ^ S4 x [0,i]
'~Zn+ 1
by
~ H ; ( f n ( X ),s,2t),
0 < t <__89
Hn(X,S,t) t
Let
F',F", and F
(f"f',{Hn}).
Then
be the maps associated with
F "~ F ' F "
in
(f',{Hn})
1.
s
t! (fi! ,~ Hn}),
and
Ho(CWSp).
Lemmas (2.2.18) - (2.2.20) yield the following proposition.
(2.2.21)
Proposition.
A permutatlon-commutatlve diagram of permutation maps
induces a commutative diagram in
Ho(CWSp).
this diagram are defined up to homotopy in
(2.2.22)
Proof of Theorem (2.2.14).
The maps and required homotopies in Ho(CWSp).
There are destabillzatlons (2.2.15) and
natural permutation classes of permutation maps D'(X ^' Y) --~X ^ u
D
The composite mappings
D(X A Y) --~X A' Y~ D'D(X ^ Y) ---~X ^ Y
and
23
DD'(X A' Y) - - ~ X A' Y permutations.
differ
from
the
The conclusion follows.
(2.2.23)
Theorem.
X
respective
>X
a:(X A Y) ^ Z c:X ^ Y
~Y
inclusions by
Q
There are natural maps in
~ X A S0
(cofinal)
Ho(CWSp),
(unit),
~ X ^ (Y A Z) ^ X
(associativlty), (commutativity),
which yield a symmetric monoidal category in the sense of S. Eilenberg and G, M, Kelly
[E -K].
Proof.
There are destabilizatlons
and natural permutation
classes of permuta-
tionmaps
DX
> X ^ S O.
~ X,
a':D((X ^ Y) ^ Z)
~X
^ (Y ^ Z),
c':D(X A Y) ----~y A X
(D
is used generically).
associated maps of
By Proposition
CW
Let the maps
spectra
(2.2.21),
permutation -commutative
DX--~ X ^ S0---~X,
((2.2.17)
a,
and
c
- (2.2.!8)).
it suffices to obtain the coherency diagrams
c
2:D2(X A
[E- K] as
diagrams of permutation maps between destabilizations.
These diagrams include statements that the above maps are isomorphisms, composite map
be the
Y) ---~X ^ Y
certain coherency conditions hold. coherently homotopy associative is
is homotopic For example,
to the identity,
that the
and that
the diagram stating that
A
is
24
D2(a')
D3(((W ^ X) ^ Y) ^ Z)
D2((W ^ X) ^ (Y ^ Z))
L
I
(a' ^ id).
D(a')
i
D2((W ^ (X ^ u
^ Z)
I
D(W ^ (x ^ (Y ^ z)))
D(a')
(id ^ a'),
D(W ^ ((X ^ Y )
^ z)
"~ w ^ (x ^ (Y ^ z)).
These diagrams are readily obtained.
(2.2.24)
Remarks.
able smashproduct on
The approach of Theorems (2.2.22) - (2.2.23) yields a suitHo(CWSp)
for studying homology and cohomology theories and
operations involving maps of spectra.
It does however ignore higher homotopies
which yield the rich structure of infinite loop spaces [May -3]. incorporate this structure in their smash products.
May and Puppe
It would be interesting to
obtain higher homotopies within the above simple framework, perhaps with a suitable operad.
See P. Malraison [Mall.
(2.2.25)
~urther properties o f
^ -
It is easy to verify that
^
commutes
with the suspension
S 1 ^ ?: Ho(CWSp)
^
) Ho(CWSp),
satisfies a Kunneth formula for stable integral homology, and
universal for pairings [Wh-l],
^
is weakly
[K-W].
We shall now define function spectra, and verify that the smash product
^
Ho(CWSp), together with
and function spectra, forms a s/amnetric monoidal closed
category in the sense of Eilenberg and Kelly [E- K].
This includes the usual
2S
exponential law.
(2.2.26)
Construction of function spectra.
Let
For a first approximation to the function spectrum spectrum
Map(X,Y)
(s-4k)n = s4n- 4k
as follows. for
n _> k,
induced by isomorphisms smash product Ho(CWSp)
^'
on
Let and
*
S -4k
X
HOM(X,Y),
be the
be
CW
spectra.
define a weak
-4k - sphere
spectrum:
otherwise; the required inclusions are
s4n - 4 k ^ S 4 ~ S4n -4k + 4 CWSp
and Y
using (2.2.13).
Choose a representative
The eventual function spectrum in
will be independent of this choice.
Define
Map(X,Y) n = CWSp[S -4n A X,Y)
with the topology induced from the compactly generated function spaces
P(IS-4n ^ X')i' S 4 A S -4n -4
where
".s-4n
(S 4
X'
is a coflnal subspectrum of
X.
The maps
is a space, the other terms are spectra) induce the
required maps
Map(X,Y)n A S 4
This yields a weak spectrum
(2.2.27) Map(X,Y),
Definition.
Map(X,Y).
Let
HOM(X,Y)
be the
CW
spectrum obtained from
see Remarks (2.2.7)(c).
We can extend show that
~Map(X,Y)n+l.
HOM
(2.2.28)
HOM
to bifunetors on
CWSp
and
Ho(CWSp).
is the required internal mapping functor.
Theorem
(exponential law) .
Ho(CWSP)(X A Y,Z) = Ho(CWSp)(X, HOM(Y,Z)).
We shall now
26
Proof.
If
X
and Y
(2.2.29)
Because a
CW
are finite spectra, it is easy to show that
CWSp(X ^' Y,Z) = CWSp(X, Map(Y,Z)).
spectrum is the colimit of its finite subspectra (the Boardman-Heller
completion, see (2.2.11) and the following discussion), CW
spectra.
(2.2.29) holds for arbitrary
This also yields an analogous formula in
there are natural weak homotopy equivalences
Ho(CWSp).
HOM(Y,Z)
Finally,
~ Map(Y,Z)
jections of mapping cylinders and the natural transformation
(from pro-
R o Sin--+id).
Thus
Ho(CWSp)(X, Map(Y,Z)) ~ Ho(CWSp) (X, HOM(Y,Z)),
by [Adams -i, Theorem 3.4].
(2.2.30)
Corollary.
The conclusion follows.
HOM( , ) = Ho(CWSp)(
This follows from the definition Ho(CWSp)
is normalized
(2.2.31)
Theorem.
D
, ).
T 0 = Ho(CWSp)(S 0, ).
This also shows that
[E -K, p. 491].
Ho(CWSp),
together with the above structure, is a symmet-
ric monoidal closed category.
The remaining coherence conditions fied.
[E - K, p. 491, Theorem 5.5] are easily veri-
Their precise statement and proof is omitted.
(2.2.32)
Remarks.
Adams defines an internal mapping functor
ing to Brown's Theorem (see [Adams - i]).
HOM
by appeal-
This approach also yields the above
theorem.
This concludes our formulation of the category of
CW
spectra.
We shall now
briefly discuss simpllclal spectra and the equivalence of homotopy categories. the end of this section we shall use
CW
At
spectra to discuss homology and cohomology
theories and operations, following [Adams -1,3].
27
(2.2.33) funetor
Definition [Kan -i]; we follow [May- i].
E:SS,---~SS,
is defined on objects as follows.
simplicial set with basepoint all symbols
(i,x)
identifications
where
i
* .
Then
(EX) 0 = *0'
is a positive integer and
(i, *n_i ) = *n
XK).
basepoint in
The simplicial suspension
(*k
Let
X
and
be a pointed
(EX) n
x ~ Xn_ i,
consists of subject to the
denotes the appropriate degeneracy of the
Face and degeneracy maps are defined as follows:
So(i,x) = (i+l,x)
Si+l(l,x) = (l,six)
~0 (l'x) = *n'
x ~ Xn
~i (l'x) = *0'
x e X0
%i+l(l,x) = (l,~ix),
x ~ Xn,
n > O,
and by the simplicial identities (2.2.1).
Then
EX
is a pointed simplicial set with one non-degenerate
every non-degenerate
(n -i) -simplex
of X
reduced suspension of the realization of the realization
(2.2.34) prespectrum
except the basepoint. E RX
Further, the
is canonically homeomorphic to
[Kan- i], compare Definitions (2.2.5).
A simplicial
consists of a sequence of pointed simplicial sets
with inclusions
EX(n) --->X(n +i).
extends
El(n).
Let
X(n),
A map of simplicial prespectra
consists of a sequence of pointed simplicial maps f(n +i)
simplex for
REX.
Definitions X
X,
n
Ps
f(n):X(n)
> Y(n)
together f:X---~Y such that
denote the category of simplicial prespectra.
28
We could "complete" the category of slmplicial prespectra as in Definitions (2.2.6).
However, Kan introduced a conceptually simpler completion which essen-
tially replaces a simplicial prespectrum by its associated ~ - spectrum (see (2.2.7)(b)).
An
n +k
simplex of
(n + k +i) - simplex of spectrum
SX
X(n)
X(n + i)
associated to
X
(element of under
X(n)n+k)
E.
corresponds to an
Thus Kan defined the simplicial
by taking as stable k -simplices the pointed
(*)
sets
S~
= u X(n)n+k,
k
~
Z,
n
+ k ~ O.
There are induced face and degeneracy maps of pointed sets
di SXk+ 1 --~ SX k si:s
for
1
which satisfy
i > 0
i)
-
the usual simpllcial identities (2.2.1)
and
•
the local finiteness condition:for every simplex there is an integer d.o = *
for
n
o
in SX,
(depending upon o ) such that
i > n.
1
(2.2.35) (*)
sets
~
Definitions
[Kan- i].
(the ~ - slmplices of
A simplieial spectrum X),
k E Z ,
degeneracy maps which satisfy the above conditions.
X
consists of pointed
together with face and A map of slmplicial spectra
29
f:X---~Y
consists of a sequence of maps of pointed sets
commute with face and degeneracy maps.
Let
Sp
{fk:~
~ Yk }
which
be the category o_~fslmplicial
spectra.
The spectrum construction
S
above extends to a functor
admits an adjoint prespectrum functor PX
is defined by letting
with
d.o = * l
for
PX(n)j
i > j.
The categories
Sp
P:Sp --->Ps,
and SS
S
on a simplicial spectrum
consist of those
Further,
S:Ps ---~Sp.
(j -n)
simplices
o
X, of Y
SP = identity:Sp --~Sp.
enjoy many similar properties.
In particular,
there is a Kan extension condition for simplicial spectra, and the homotopy groups of a Kan spectrum admit a combinatorial definition [Kan -i].
More generally,
K. S, Brown proved the following theorem.
(2.2.36)
Theorem
[Brown].
The category of simplicial spectra admits a
natural closed model structure in the sense of D. G. Quillen [Q -i], see w
We shall sketch an independent proof in the spirit of Quillen's proof [Q -i, w
that
SS
is a closed model category.
standard simplices which do not exist within BSp
Sp,
The first task is to define but only in a larger category
of big simplicial spectra.
The definition of a big simplicial spectrum and the category to the definition of simplicial spectra and
Sp
BSp
is analogous
(2.2.35) except that there is n__0_o
local finiteness restriction (li).
Sp
is a full subcategory of
an adjoint
T ; on objects in
BSp.
BSp,
Also, the inclusion T~
J:Sp --~BSp
consists of those simplices in
admits ~
with
30
almost all faces at the basepoint extend the functor
P:Sp---~Ps
(condition to
Following V. K. A. M. Gugenheim
BSp;
(ii) in (2.2.35)). then
Alternatively,
T = SP.
[Gug, p. 36], a simplicial operator
composite of face and degeneracy maps.
Each simplicial operator
~
~
is a
has a unique
standard representative.
~ = s .-- s s d d -'" d mi m 2 mI nI n2 nj
i
with
0 < m I < m 2 < "'" < m i
is defined to be
(2.2.37)
and
0 _5_ n I < n 2 < "'" < nj .
i -j.
Definition.
The standard k - simplex
spectrum with one non-degenerate
k - simplex
Anm= where
~
{r
~k
Ak
disjoint basepoint
U {*},
Proposition.
finiteness
spectrum of
with the maps
Remarks.
For
Definition. Ak
D
the maps
to Condition
(ii).
The boundary of
generated by the faces
spectrum are in
Ak--->Y.
in Sp,
Y
condition analogous
(2.2.40)
together with a
The k - slmplices of a big simplicial
As in [G -Z, Theorem i].
(2.2.39)
m -k,
* .
one-to-one correspondence
Proof.
is the big simpllcial
and whose m - simplices are given by
ranges over all simplicial operators of height
(2.2.38)
The height of
di~ k
Ak, aAk, of A k,
Ak--->Y
satisfy a local
is the big simplicial i ~ 0.
The horn
sub-
V k's
31
is the big simplicial subspectrum of
3A k
generated by the faces
We discuss the extension condition for
BSp
and
Sp,
di~ k
for
i #
and the homotopy groups
of a simplicial spectrum.
(2.2.41) spectrum
X
Definition
(Compare [Q -i,
11.32 for
is said to be Ken if every map
(2.2.42)
Remarks.
SS).
V k'g --+X
A big slmplicial can be extended to
A k,
Ken [Ken -i, Def. 7.3] said that a simplicial spectrum
satisfied the extension condition if for each
n,
the simplicial set
satisfies the Ken extension condition, see, e.g.,~ay-l~)ef. 1.3].
X
PX(n)
It is easy to
check that the definitions are equivalent.
We shall now define the homotopy groups of a Kan simplicial spectrum, and extend the definition to
(2.2.43) o
and o'
Definition.
of X
Sp.
Let
X
be a Ken simplicial spectrum.
(Compare, e.g.,
are homotopic, denoted
[May - i, Def. 3.1]). x ~ x',
Two k - simplices
if for all
i,
d.o = d.o', 1
and if there exists a
(k +i) -simplex
T
with
d 9 = o,
dl~ ~ o',
1
and
O
di~ = diSoO = diSoO'
(2.2.44)
for
Proposition.
spectrum, then
i ~ 2.
(Compare [May -i, Proposition 3.2]).
If
is an equivalence relation on the k - simplices of
X X,
is a Ken for all
k.
Proof.
(2.2.45) spectrum.
As in the proof of [May -i, Proposition 3.2].
Definition. Let
~_
(Compare [May-l,
Def. 3.6]).
Let
denote the set of all of the k - s i m p l i c e s
X
be a Kan a
of X
which
32
satisfy
dlo = *
for all
i.
Define
~k(X) = ~ / ~
By imitating the discussion in [May -i, w
(2.2.46)
Proposition.
~k(X)
,
K
and ~ (K) m
for
we see that
is an abelian group.
Finally, for a pointed simplicial set dition, define
9
(K,*)
D
which satisfies the extension con-
as above for
m > 0.
Wm(K)
is a group
m
m > i.
(2.2.47)
Proof. since
Proposition.
~k(X) = ~k+nPX(n)
A k - simplex
dlo = *
o in ~
for all
i,
for all
can be realized in
hence for
k +n ~i.
PX(n)
for
n+k
> 0
n + k ~ 0, ~
c (PX(n)) k
Similarly, for
"
n +k sO,
(PX(n))
~
c ~
.
Further the simplices needed to define the equivalence relation in realized in (we need
PX(n +k) n +k ~ 1
for
for
n + k ~ 0,
~n+kPS(n)
Hence our definition of
~k(X)
and conversely.
may be
The conclusion follows
to be a group).
agrees with that of Kan [Kan- i, w
The following proposition is essentially contained in [Kan -i, w
We give
an explicit proof because of the variety of equivalent definitions of weak equivalence in
SS
(2.2.49)
(the equivalent conditions in [Q -i, Proposition w
Proposition.
A map
f:X--~y
in Sp
is a weak equivalence if and
only if for
Proof. PX(n)
n h 0,
Let
the maps
f:X---~Y
Pf(n):PX(n)
~ PY(n)
be a weak equivalence.
and for any basepoint
x
in PX(n),
are weak equivalences.
We shall show that for any
and for any
m ~ 0,
the induced
map
w (PY(n),y), n
Pf(n),:~m(PX(n),x)
where
y = Pf(n)(x),
is an isomorphism.
connected subsimplicial set of (the basepoint of
PX(n + i),
To see this, since
EPX(n)
is a
and contains the standard basepoint
X),
m(PX(n)'x) ~ ~m+l(PX(n+l),Ex)
-~ ~m+l(PX(n +i) ,*)
~m+l(PY(n +i),*)
~n+l(PY(n+
by Proposition 4.7)
1),Ey)
! ~ (PY(n),y); m
further, the isomorphisms induced by changing basepoints may be chosen so that the composite is
Pf(n),.
Hence the map on realizations
R(Pf(n)) : R(PX(n))
is a weak equivalence.
~ R(PY)(n))
By [Q -i, Proposition w
so is the map
Proof of the converse follows immediately from Proposition (2.2.47).
Pf(n).
D
34
(2.2.49)
Remarks.
Relative homotopy groups, the homotopy exact sequence for
a pair, the fibre of a flbration, and the homotopy exact sequence of a flbration for Sp,
as in [Kan -3, w167
See, e.g.,
may be obtained by imitating their developments in
[May -i, w167
Cofibrations and fibrations in in
SS
SS.
[Kan-2],
(2.2.50)
Sp
see e.g., [May -i],
Definitions.
are defined analogously with the definitions [Q-I,
w
Cofibrations are injective maps.
A map
E-->B
of
simplicial spectra is a (Kan) fibratlon if given any commutative solid-arrow diagram in
BSp
of the form
~s
~E / .1
#, t /e" / I k
there exists a filler
:B
f.
We can now prove Theorem (2.2.36), that
imitating Quillen's proof that
SS
Sp
is a closed model category, by
is a closed model category [Q -i, w
Details are omitted.
We shall now summarize Adams'
formulation of the foundations of generalized
homology and eohomology theories; see [Adams -1,3].
Let
h,
be a non-negatively
graded generalized homology theory defined on finite complexes and
h
be the dual
cohomology theory.
(2.2.51)
that
Brown's Theorem [Bro].
h4n(x) ~ Ho(CW,)(X,E4n)
(2.2.52) above,
Theorem
There is a
CW
spectrum
for all finite pointed
(G. W. Whitehead
[Wh-2]).
With
CW
E = {E } n
complexes
h,
and
such
X.
E = {En}
as
35
~(X)
= col/m n ___> Ho(CW) IS4 n + k,x A En) .
We use (2.2.51) and (2.2.52) to define
h
and
h,
CW
on all
spectra:
hk(X ) ~ Ho(CWSp)(sk,x A E)
(2.2.53)
E ~k(X ^ E)
hk(x) ~ Ho(CWSp)(X A s-k,E).
(2.2.54)
Alexander and Spanier-Whitehead duality.
hedron linearly embedded in
Sn.
Let
K
be a compact poly-
By Alexander duality there are isomorphisms
HP(K) ~ Hn_p_l(Sn\ K).
E. Spanier and G. W. Whitehead extended Alexander duality to state that determines the stable homotopy type of
Sn \ K
and that
Sn \ K
K
has the homotopy
type of a compact polyhedron.
Spanier introduced the following formulation of duality. compact polyhedra disJointly embedded in K
to L
with
Regarding
Sn
~(0) ~ K,
m(1) E L,
and
as the compactification of
yields disjoint embeddlngs of
(2.2.55)
Sn.
K
~(0,I) Rn
and L
Choose a
PL
path
disjoint from
with
in R n.
~:K • L
Let
~(89
K
and L ~
be
from
K u L.
as the "point at
~,"
Let
>S n-I
be the map
(2.2.56)
p(k,s
=
It is easy to check that the restrictions
~
k-s
I1~-~11
.
~I~(0) x L
and
~[K x ~(i)
are null-
38
homotopic.
Let
~(0)
Then (2.2.37) yields a
and
~(i)
be the basepoints of
and L
respectively.
map
(2.2.55)
u:K A L
Regarding
K
K, L, and
Sn-I
>S n-I
as spectra, and taking the adjoint of
p
in
(2.2.57), yields a map
~,:K
(2.2.58)
~HOM (L,sn-l).
Spanier proved that if the inclusion
L--->S n \ K
~,
K
is a stable homotopy equivalence.
and (2.3.58).
S n-I \ K
is called the
Dn_ID_I
(2.2.60)
play symmetric roles in (2.2.57)
(n -i) -dual
of
K:
Dn_IK ~ s n - l \ K.
(2.2.59)
Then,
and L
is a homotopy equivalence, then
=
- id.
Finally, the natural "composition"
~ HOM
Dn_l K A HOM(Sn-I,E)
(K,S n-l) A HOM (sn-l,E)
>~OM (K,E)
is a stable homotopy equivalence.
(2.2.61)
Definition.
The functional dual of a
CW
spectrum
X
is given by
DX E HOM (X,S0).
There is a natural map X A DX = S A H0M (X,S 0) finite.
X---~D2X SO)
(take the adjoint of the evaluation map
which is a stable homotopy equivalence if
X
is
37
We shall now discuss generalized homology theories represented by ring spectra~ following Adams [Adams -3, especially pp. 60-68].
(2.2.627
Definition.
multiplication map m
A ring spectrum consists of a
m:E ^ E--->E,
and a unit map
be homotopy associative and commutative and that
CW
spectrum
i:S0-->E. i
E,
a
We require that
be a homotopy unit.
J. P. May has gone considerably farther in studying the higher homotopies associated with ring spectra [May -2,3].
(2.2.63)
~
(see (2.2.7) for descriptions).
a)
S 0.
b)
K(R),
c)
BU, BO, ere; the multiplication is induced from the tensor product
where
R
is a ring.
of vector bundles. d)
MU, MO, etc.; the multiplication is induced from the Whitney sum of vector bundles.
e)
bu, bo, etc.; connected versions of
BU,
BO,
etc.;
see
D, W. Anderson [An-l,2].
Module spectra are defined analogously with (2.2.62). over
S~ .
P. E, Conner and E, E. Floyd [Con -FI] showed that
spectrum over
(2.2.64) spectrum
All spectra are modules
E~
E,(S O) = ~,(E)
BU
is a module
MU.
Let
E,
i.e.,
be a generalized homology theory represented by a ring E,(X) = ~,(X ^ E).
The coefficient ring of
with multiplication defined by
E
is
38
m, ~,(Z)
E,(X)
is a right
|
=,(E)
~ ~,(Z ^ E)
E,(S 0) -module
E,(X)
|
. ~,(E).
under the map
Z,(S ~
= ~,(X
^
E)
|
~,(E)
- - ~ , ( X ^ E A E) - - ~ , ( X ^ E).
(2.2.65)
Homology operations.
E,(E)
is a two-sided
E,(S 0) - module,
and
the two actions differ by the canonical involution c:E,(E) = ~,(E A E)
~,(E
A E) = E,(E)
shall now require that the
0 E,(E)
and
~,(s ) right
Then
E,(E)
E,(S 0) - modules
S0.K(Zp).
BO.
E,(E)
BU.
E,(E)
MU.
from right
E,(S O)
R,(S 0) - modules
This requirement holds for at least MSp.
and
though not for
becomes a Hopf algebra over
r
~
E,(E)
K(Z).
E,(S0).
The product map
> E,(E)
is the composite
~,(E
A
E)
~
We
be flat. that is. that the functors
be exact.
MO.
induced by interchanging factors.
~,(E
A
E) ~ , ( E
A
E
A
E
A
E)
(id ^ switch ^ id), > ~,(E ^ E ^ E A E) (m
^
m), . ~,(E
^
E).
to
30
Also,
~
induces a p r o d u c t over
E,(S O) (easy).
The coproduct (or d i a g o n a l map)
= ~E:E,(E)
9 E,(E)
~) E,(E)
is a special case of the coaction map
~X:Z,(x)
~Z,(X)
|
~,(E).
The coaetion map is defined as follows:
~,(X A E) = ~,(X ^ S 0 ^ E)
(id ^ i ^ i d ) , 9
~,(X h
E ^ E)
*~--w,(X ^ Z)
|
(The latter map is an isomorphism because
~,(E)~.(E ^ Z)
w,(E)
is flat [Adams -3, p. 68,
Lemma i]).
It
o f the d u a l o f t h e
i s now e a s y t o e x t e n d t h e t h e o r y o f c o a c t i o n
Sreenrod algebra on
(2.2.6~)
H,( ,Zp)
to a theory of homology "operations" for
Cohomolo~y and eohomoloF~v operations, are defined dually.
becomes a module over
E,(S O)
EP(x)
@
mod-p
as follows:
Eq(s O) ~ IX ^ s - P , E ]
(~
[sq,~]
; ix ^ S -p ^ sq,E ^ E]
IX ^ s-P+q,E] = EP-q(x).
'
E,.
E (X)
40
is also a Hopf algebra, with the product
E*(E)
E (E) ~
E*(E)
~E*(E)
induced by the composition
[E ^ sP,E]
~
[E ^ Sq,E]
>
[E ^ S p ^ S q, E ^ S q]
>
[E ^ S p a Sq,E]
~
[E ^ Sq,E]
= [E ^ SP+q,Ei.
The (right) action of
* E (E)
* E (X)
on
is defined by a similar composition.
The power of the above very general theory is easily demonstrated ing simple proof
(due to Adams
BU (X) = 0.
Recall that
X ^ S p --->BU
may be factored as follows:
(2.2.67)
BU
[Adams-4])
that if
MU,(X) = 0,
is a module spectrum over
X ^ $P
MU.
in the follow-
then Each map
~ X A Sp ^ S O
~xAsP
AMU
-~BU ^ M U
BU 9
But
~,(X A S p ^ MU) -~ ~,(X A MU A S p) ! ~,_p(X A MU) = MU,_p(X)
X A S p ^ MU "- * Hence,
by the Whitehead Theorem for
the composite map (2.2.67)
CW
is null-homotoplc,
spectra
= 0.
(see e.g.,
as required.
Thus [Adams-l]).
41
w
Model categories. We shall describe the basic properties
associated homotopy categories; [Q-I,
w167
of closed model categories and their
this theory is due to D. G. Quillen
1.5].
(2.3.1)
Definition.
An ordered pair of maps
(i,p)
liftin~ property if given any solid-arrow commutative
A
t /
/
I
t
i f
X--
(2.3.2)
~B
f.
Definition.
A closed model category consists of a category
together with three classes of maps in and weak equivalences
MO.
C
MI.
If a map
C,
called the fibrations,
is closed under finite colimits and limits. i
is a cofibration,
pair
i
(i,p)
Any map
f
or p
a map
i
is a fibration,
is a weak equivalence,
then the
has the lifting property. may be ~actored as
f = pi
cofihratlon and a weak equivalence or
p
is a cofibration and
p
and
where p
i
is a
is a fibratlon,
is a fibration and a weak
equivalence. MS.
Fibrations
(resp. cofibrations)
and base change (pullbacks)
are stable under composition
(resp., cobase change
Any isomorphism is a fibration and a cofibration.
C
cofibrations,
satisfying the following axioms.
and either
M2.
diagram
~Y .../1
there exists a filler
is said to have the
(pushouts)).
42
M4.
The base extension
(resp., cobase extension)
is both a fibration
(resp., cofibration)
of a map which
and a weak equivalence
is a weak equivalence. MS.
Let f
X
be a diagram in
C.
g
'~Y
~Z
If any two of the maps
f,g ,
are weak equivalences, then so is the third.
and
gf
Any isomorphism is
a weak equivalence.
M6a.
A map
p
is a fibration if and only if for all maps
are cofibrations
and weak equivalences,
the pair
i
which
(i,p)
has
the lifting property. M6b.
A map
i
is a cofibration
if and only if for all maps
are fibrations and weak equivalences,
the pair
p
(i,p)
which has
the lifting property. M6c.
A map
f
is a weak equivalence
for all cofibrations and
(v,p)
i
if and only if
and fibrations
p,
f = uv
where
the pairs
(i,u)
have the lifting property.
Observe that Axioms M5 and M6 imply Axioms
MI, M3, and M4;
hence to show that
a given category is a closed model category it suffices to verify Axioms M_5 and M6.
(2.3.4)
We shall want the following technical definitions.
Definitions.
A map which is both a fibration
and a weak equivalence
is called a trivial fihration
The initial object of
C
shall be denoted
objects exist by Axiom M0). natural map cofibration.
M0, M2,
X--~*
An object
is a fibration;
(resp., cofibration)
(resp., trivial cofibration).
~ ; the terminal object X
in C
cofibrant
*
(these
is called fibrant if the if the natural map
~ --~X
is a
43
(2.3.4) object
Definition.
X ~
[0,i]
Let
X E C.
(i0,il)
pl 0 = pi I = id X. X~
i
for
product of
i 0 + iI
'--x|
consists of an
[o,i] P~x
is a cofibratlon, the map
We shall frequently write
il(X ). X
X
and a commutative diagram
x_[ix where the map
A cylinder object for
Caution:
with an object
not depend functorially upon
in general, [0,i];
X | X |
p
is a weak equivalence, end
0
for
[0,i]
i0(X)
and
is not the "tensor"
in fact, in general,
X |
[0,i]
need
X.
We may use cylinder objects to form mapping cylinders with the usual properties. For example, we have the following.
(2.3.5~
Proposition.
Let
suitable cylinder objects so that
f:X---~y f
be a cofibration.
Then there exist
induces a trivial cofibration
Y • 0 u X (~) [ 0 , i ]
~-Y ~
[0,i],
and a cofibration
(2.3.6)
If
f
t ion,
Y |
u x @
[0,i] u Y
|
'-Y |
[0,1].
is a trivial cofibration, the induced map (2.3.6) is also a trivial cofibra-
44
Proof.
Consider the commutative diagram
i0 + i1
x | o u x |
i
~
>x |
[o,i]
,Y |
0 u X |
{ I I
JO+Jl
Y @)OuY ~ i
9
[0,i] u Y |
\
1
\,
\,
k
idy +
Y |
\N
[0,11
"
\
q-.
\ \
\.
\
in which the subdiagram i0 + iI X ~
0 u X ~
is a cylinder object for upper left square, and The map
JO + Jl
L
X, g
Y ~
0 u X ~
[0,i]
[0,I] u Y ~
Y,
f
1
is the pushout of the fp
and i ~
Factor
g
as
qk
where
k
+ i ~l .
0 u Y ~
1
induces cofibrations
kJo+kJl;Y
~
[0,i]
i0 + il,
is a cofibration and
We obtain a suitable cylinder
namely
Y ~
so that
*-X
is the pushout (cobase extension) of the cofibration
is a trivial coflhration (dotted arrows above).
object for
P
is the map induced by the maps
hence is itself a cofibration. q
~X ~
q
;Y,
4S
Y |
0 u X |
Y (~) 0 u X |
>Y
[0,1] --
[0,1] u Y |
|
; Y ~
1
[o,i],
[0,i].
The remaining assertions are easily checked by applying Axiom MS; details are omitted.
We shall discuss cocylinder objects (dual to cylinder objects) in
C,
and
loop and suspension functors as well as the induced cofibration and fibratlon sequences in w
below.
In these notes we shall always assume that our closed model categories
C
satisfy the following niceness condition.
Condition N: NI.
Each cofibration is a pushout of a cofibration of cofibrant objects.
N2.
Each fibration is a pullback of a fibration of fibrant objects.
N3.
N4.
At least one of the following statements hold: N3__aa.
All objects are cofibrant.
N3b.
All objects are fibrant.
There exist functorial cylinder objects, denoted by
-~[0,i]
with
io(- ) = - G O
and
il(-) = - ~ i .
The following closed model categories satisfy Condition N.
SS; (D. M. Kan [Kan -3],
all objects are cofibrant, Condition due to J. C. Moore, see [Q -i, w X ~
[0,i] = X x A I,
Since
[Kan -4], see D. Quillen [Q- i, w N(1) For
the usual product.
is trivial. N(3),
let
N(2)
is
46
Top;
the category of topological spaces with the following structure: cofibrations and fibrations are defined by the homotopy-extension and covering-homotopy properties, respectively; weak equivalences are ordinary homotopy equivalences. Condition
CG;
N
This is due to A. Str6m [Str].
is clear.
the category of compactly generated spaces, with a similar structure. See N. E. Steenrod [St -3]; also,
Sing;
[Has -3].
the category of topological spaces with the following singular structure: of
CW
cofibrations are pushouts of inclusions of w
complexes, fibrations are Serre fibrations, weak equivalences
are weak homotopy equivalences
[Q -i,
w
Again, Condition
N
is clear.
SSG (SSAG);
Simplicial groups (resp., simplicial abelian groups)
[Q -I, w
x
•
Here let
[o,1]
=
F(X
•
[O,I])/F(X
• O)
where the products are taken in
SS
abelian) simplicial group functor. Condition
Sp;
N
for
- X
• O,
and F Condition
Sp
• l)
- X • l,
is the free (resp., free N
follows from
SS.
D, M, Kan's simplicial spectra [Kan -i]. proved that
F(X
K. Brown [Brown] first
is a closed model category with a closed model
structure similar to that on follows from condition
N
for
SS.
See w
Condition
N
SS.
We shall now describe the homotopy theory of a closed model category satisfies condition
(2.3.7)
C
which
N.
Definition
[Q- i].
quotient category obtained from
The homotopy category of C
C,
Ho(C),
by inverting all weak equivalences.
is the
47
Quillen proved the following.
(2.3.8)
Proposition
equivalence in
Ho(C)
[ Q - I , Prop. 1.5.1].
if and only if
f
A map
f
in C
becomes an
is a weak equivalence in
C.
The following homotopy theory is required for the proof.
(2.3.9)
Definition.
f,g:X---->Y
~Y
(2.3.10)
with
Definition.
fibrant objects in
(2.3.11)
with
X
is cofibrant and
will be called homotopic (denoted
H:X • [0,i]
f:X--~Y
If
gf = id x
Let
Ccf
and
f = g)
HIXI •
is fibrant, maps if there is a map
= g"
Compare
[Q -i,
w
denote the full subcategory of cofibrant,
C .
Proposition
in Ccf
HIXI x 0 = f
Y
[Q -i, Lemma 1.5.1 and its dual].
A map
is a weak equivalence if sndunly if there is a map and
E:Y-'~X
fg = ~ .
The proof is analogous to that of [Q -i, Lemma 1.5.1], and is omitted.
Proof of Proposition (2.3.8). invertible in
Ho(C),
Given a map
f:X--->Y
I 1
f
' Y
X'
x . . . . .
X'
and Y'
which is
form a commutative diagram
X
where
in C
are cofibrant,
I 1
Y'
[f"k
X"
_
_
.~,,
and Y"
,
are both cofibrant and fibrant,
48
and all vertical maps are weak equivalence. [f"] f"
in Ho(C)
by a map
f"
is a weak equivalence.
w
Simplicial
in C.
Then use the axioms to realize the map
Finally, Proposition
See [Q -i,
w
for details.
SS,
and their generalization
cept df simplicial closed model category given by Quillen
The product in
SS
(or internal mapping functor)
[Q -I,
This product is coadjoint
> An ,
of the adjoint pair
The functors
(i)
~
0 ~ i in.
See e.g.,
[May-l] ~
or [ Q - I ] .
(x, HOM)
we shall write
and
satisfy the following properties.
HOM
for
x .
There is an associative composition HOM (X,Y) x HOM (Y,Z) X, Y, and Z functors
in C,
such that for and
g
u
~HOM
(X,Z),
for all
and a natural isomorphism of
SS(X,Y)
> HOM (X,Y)0,
in SS(X,Y),f
in H O M (W,X),
together with
to the "function
HOM (X,Y) = {HOM (X,Y) n = SS(X x An,y)},
together with the face and degeneracy maps induced form the maps s i :d n+l
to the con-
w167
X x y = {(X x Y)n = Xn x yn} ,
is given by
the induced face and degeneracy maps.
and
implies that
closed model categories.
We shall discuss function spaces in
space"
(2.3.11)
in HOM (u
n,
di:A n-I
~A n
In the context
49
f o (s0)n5 = HOM (u,Z)n(f),
(SO)n f o g
=
and
HOM (W,U)n(g)
r~ SS(X,Y)
for all
(ii)
Y
9 HOM (X,Y)0,
in C.
There are natural maps
u:Y
~ H O M (X,X ~
Y)
which
induce isomorphisms (enriched adJunction [E -K])
HOM (X ~ Y,Z) ----~HOM (Y, H0M (X,Z)),
(iii)
for all
X
and Z
in C.
For all
Y
in C
there are natural maps
B:Y-"~HOM
(HOM (Y,Z),Z)
HOM (x, HOM (Y,Z)) for all
(iv)
X
and Z
which induce isomorphisms
~HOM (Y, H0M (X,Z)),
in C.
There are natural isomorphisms
HOM (*,X) ~
for all
X
X,
in C.
Consequently,
(v)
The composition maps in (i) and
SS
are compatible; i.e.,
the following diagram commutes for all
X,Y,Z
in SS:
50
HOM (X,Y) 0 x HOM (Y,Z) 0
~ HOM (X,Z) 0
ss(x,Y) x ss(Y,Z)
(vl)
~
> ss(x,z) .
is coherently associative and co-~utatlve,
with
*
as coherent unit.
Function spaces and products example,
define
~
on SS,
in
by
CG
have similar properties.
X~Y
is replaced by
- X ^ Y = X x Y/X v Y,
above except that
An
Joint basepolnt.
These ideas have been abstracted
closed symmetric monoidal category
A n*
admits a "singular
HOM (X,Y)
function space."
to be the simpllclal
An
and
HOM
as
by adjoining a dis-
in S. Eilenberg and G.M. KelIy's
[E- K].
The singular closed model structure on w
obtained from
For another
Top,
([Q -i, w
TOPsing
For
X
and Y
in Top,
define
HOM (X,Y) n = Top (X x RA n , y)
set with
see
together
with the induced face and degeneracy maps, see above.
Mere
R
denotes M/inor's
geometric realization functor
For
X
in
K
in SS,
define
X ~
[Mil - 2], see [May -i].
K ffi X x RK.
HOM (X ~
for
K
in SS,
and
X,Y
is a category
Definition C
and
Then
K, Y) = HOM (K, HOM (X,Y))
in Top.
Quillen generalized
ducing closed simplicial model cate~orles,
(2.4.1)
Top
this concept by intro-
described below.
(see [Q -i, Definition
w
together with the following structure:
A simpliclal category
51
(i)
A functor
HOM (-,-)
from
C x C
to SS,
contra-
variant in the first variable and covarlant in the second.
(ii)
X, Y, and Z
For an
in C,
maps in
SS
H0M (X,Y) x HOM (Y,Z) ---~HOM (X,Z)
called composition.
(iii)
An isomorphism of functors
C(X,Y)
where
HOM (X,Y) O,
consists of the O - simpllces of
HOM (X,Y) 0
HOM (X,Y).
These functors are required to satisfy t h e following conditions. (i)
Composition is associative.
(2)
For
u
in C(X,Y),
f
in HOM (Y,Z)n,
and
f - (s0)nQ = HOM (u,Z)n(f),
g
in HOM (W,X) n,
and
(s0)nQ o g = HOM (W,U)n(g).
(2.4.2) K
in SS,
map
u:K
Definition X ~
K
(see [Q -i, Definition w
shall denote an object of
HOM (X, X ~
K)
8:K
K, Y)
shall denote an object of
> HOM (HOM (K,X), X)
X
in C
ROM (K, HOH (X,Y)).
C
and
together with a distinguished
which induces a natural isomorphism
HOM (X ~
HOM (K,X)
C
For
together with a distinguished map
which induces a natural isomorphism
52
ROH
(2.4.3)
(Y,
Examples.
(K,X))
HOM
Clearly
SS,
is a simplicial category (see [Q -i],
HOM (K, HOM (Y,X)).
with its usual symmetric monoidal structure, [May], [E -K]).
TOPslng
(see w
is
also a slmpllclal category.
(2.4.4)
Definition
A closed simp!icial model
[Q- i, Definition w
category consists of a closed model category
C
which is also a simplicial category
satisfying the following two conditions.
SM0.
SMT.
For
X
in C
X ~
K
and
If
i:A---~X
and K
a finite simplicial set, then
HOM (K,X)
exist.
is a coflbration in
is a flbration in
HOM (X,Y)
C,
C
and
p:Y---~B
then
> HOM (A,Y) x HOM (A,B) HOM (X,B)
is a fibratlon in
SS which is trivial if either
i
or p
is trivial.
Recall that for spaces
X
and Y,
say in
CG,
[X,Y] ~ ~o(HOM ( X , Y ) ) , where with
HOM (X,Y) Y
fibrant (i.e., Kan),
[X,Y] ~ ~0(HOM (X,Y)).
in an abstract simplicial closed model category
(2.4.4) C
and Y
Similarly, for
is the usual function space.
Proposition is fibrant in
[Q -i, Proposition w C,
then
X
and Y
in SS
A similar statement holds
C.
If
X
is cofibrant in
53
Ho(C)(X,Y) a ~0(HOM (X,Y)),
the set of path components of
(2.4.6) a)
HOM (X,Y).
D
Remarks.
Our use of S • S, HOM
HOM
in three settings, on
C x C,
C x S,
and
should emphasize the analogy between the functor
of Definition and the usual "function space" (internal
mapping) functors.
b)
All of the above results have pointed analogues; replace SS
by SS,
and
An
by A
n*
,
which is obtained from
An
by adjoining a disjoint basepoint.
c)
In general the function space in
Top
or CG,
HOM (X,Y),
is not homotopy equivalent to the realization of the singular "function space"
R(HOMsing(X,Y)) E R{Top (X x RAn,y), di'si}"
For example, the latter space is always a
w
CW
complex.
Homotopy theories of pro -spaces.
In this section we shall briefly indicate the need for a "sophisticated" homotopy theory of pro -spaces.
M. Artin and B. Mazur took
pro -Ho(Top)
to be the
homotopy theory of
pro -Top.
for some purposes.
For example, Quillen [Q -i, C. II, p. 0.3] observed that the
category
pro -Ho(Top)
pro - Top. to call maps
Unfortunately, this point of view is inadequate
was not the homotopy category of a model structure on
One next attempts to define homotopy ~lobally in f,g:{X i} --~{Yj}
pro- Top,
homotopic if there is a homotopy
that is,
54
H:{Xi} x [0,i] ~ {X i • [0,i]}
~ {Yj}
from
f
to g.
This notion is stronger
than the Artin-Mazur notion which would identify two level maps
if there were homotopies
H .
For example, let
n
Hn:~n = gn
D
S1 z 2
S1
S1 ~
*",
is the degree two map
z t
2 ~z .
many global homotopy classes of maps from a point to Z ~2
...}
such classes).
Then, there is a unique
D,
but there are uncountably D
0
I////////i!
JJJ//J,'JJJJJH X
(more precisely,
The following example shows that the
notion of global homotopy is also too naive.
o
~ Yn}
without any Coherence criteria among the
Artln-Mazur homotopy class of maps from a point to
lim I {Z ~2
fn,gn
denote the inverse system S 1 <2
where
{Xn
P
-> Y
55
here
X = { X n ~ (S 1 v [ 0 , |
Y = {Yn s $1 x {0,1} u [ 0 , 1 ] } .
x {0,1}
u [n,|
The map
x [0,1]},
p = {pn}:X----~Y
e q u i v a l e n c e , b u t t h e r e i s no homotopy i n v e r s e t o t h e b o n d i n g maps o f t h e t o w e r s
X and Y
shall define
p
i s l e v e l l y a homotopy
i n pro - T o p .
are fibrations,
g l o b a l homotopy t u r n s o u t t o be t h e " r i g h t " n o t i o n . Ho(pro - T o p ) ,
and
If,
however,
then the notion of
The " r i g h t " homotopy c a t e g o r y ,
i s d e f i n e d by f o r m a l l y i n v e r t i n g l e v e l homotopy e q u i v a l e n c e s . Ho(pro - T o p )
in w
We
w
w
THE MODEL STRUCTURE ON PRO - SPACES
Introduction.
In this chapter we shall associate to a closed model category fies condition
N
(w
a natural closed model structure on
C
which satis-
pro - C.
This
chapter is organized as fellows.
In w
we discuss the homotopy theory of
is a cofinite
model structure on
(Theorem (3.2.2)) which is natural in the following sense
structure.
= b).
J
(a < b
(Theorem (3.2.4)).
b < a ~a
where
strongly directed set CJ
and
C J,
The constant diagram functor
The inverse limit functor
lim:C J
We shall develop a closed
C--~C J C
preserves the model
preserves fibrations and
trivial fibrations.
In w CJ
to
we shall extend the closed model structure from the level categories
pro - C
(Theorem (3.3.3)) with the same naturality properties as our
closed model structure on
CJ
(Theorem (3.3.4)).
model structure on the full subcategory on
pro - C
w
We also obtain a natural closed
tow -C c pro -C.
Simplicial structures
are discussed in w
is concerned with suspension and loop functors, and cofibration and fibra-
tion sequences.
D. Quillen
[Q - i, w167
developed a general theory of suspen-
sion and loop functors, and cofibration and fibratlon sequences in the homotopy category of an abstract closed model category. context of
Ho(pro - C).
We shall sketch this theory in the
We shall show that an inverse system of fibrations over
57
C
is equivalent in
In w
Ho(pro- C)
to a short fibration sequence.
Maps (pro -C)
we consider the category
(C, pro -C)
A--~X
whose objects are maps
(in
and a full subcategory pro-C)
with
X
stable in
pro -C.
We develop useful (see w167 Ho(tow-Top)
geometric models of
Ho(Top, tow- Top)
and
in w
We shall compare our closed model structure to those of A. K. Bousfield and D. M. Kan [B-K, p. 314] and J. Grossman [Gros-l]
in Remarks (3.2.5).
The above theory of pro - spaces admits an evident dualization to direct systems (in~ -spaces).
w
We shall briefly sketch this theory in w
The homotopy theory of Let
C
J ( = {j})
C J.
be a closed model category which satisfies Condition N (w be a cofinite strongly directed set.
a natural closed model structure from category
Ho(C J)
(3.2.1)
C;
We shall show that
A map
f:X--->Y
cofibrations (resp., weak equivalences)
in CJ
respect to all maps
inherits
(see w
Definitions.
f
CJ
this will yield the required homotopy
( = {fj :Xj -'-~Yj})
cofibration (resp., weak equivalence) if for all
A map
Let
j
in J,
in C J
the maps
is a f. J
are
in C.
is a fihration if it has the rlght-lifting-property with i
which are both cofibrations and weak equivalences.
The main result of this section is the following.
S8
(3.2.2)
Theorem.
C J,
together with the above structure, is a closed model
category.
map
A map
f:X--->Y
in CJ
qj
in the diagram
is a fibration if for each
Xj
.
j
in J,
the induced
.
"'\\
(3.2.3)
> limk < j ~
Pj
', f.'\ \
\
l~mk < jfk
Y. 3
(Pj
-~ i ~ k < jYk
is the pullback) is a fibration.
The special case tow-C
J ffiN
is used to obtain a closed model structure in
in w
(3.2.4)
Theorem.
The constant diagram functor
tions, fibrations, and weak equivalences.
C---~C J
preserves cofibra-
The inverse limit functor
lim:C J "-~C
preserves fibrations, and trivial flbrations (maps which are both fibrations and weak equivalences).
Proof.
I ~ e d i a t e from Definitions (3.2.1).
R
(3.2.5)
Remarks.
model structure on
Bousfield and Kan
CJ
defining eoflbrations
[B -K,
by defining fibrations by the appropriate
structure has the disadvantage tlons and weak equivalences
p. 314] defined a different closed
and weak equivalences
lifting property.
that most of Theorem
(3.2.4) is false:
structure
only fibra-
are preserved by the constant diagram functor; none of
homotopy inverse limit functor
Bousfield-Kan
and
The Bousfleld-Kan
the model structure is preserved by the inverse limit functor.
evident the applications
degreewise,
(w
Consequently,
our
is simpler than theirs; this simplicity makes
to homological
algebra in w167
is natural on direct systems,
see also below.
see w
especially,
The
(3.8.1)-
(3.8.4).
J. Grossman
[Gros -i] also introduced a closed model structure on the category
of towers of simpllclal sets. inverts ~ - i s o m o r p h l s m s
His structure is weaker than ours; essentially,
in the sense of [ A - M ] .
See (5.4.4)-(5.4.5)
he
for the
definition of ~ -isomorphism.
Our definition of fibration was motivated by the definition of a cofibration pairs
(for the inclusion
that the induced map of a cofibratlon of a)
(X,A) --->(Y,B)
X uA B--->y CW
spectra
be a cofibration),
(see [Vogt -2]).
one usually asks
and the analogous definition
Also:
Our definition of flbration is consistent with the definition of a flasque p r o - g r o u p
b)
to be a cofibration
(see w167
4.8).
The associated definition of coflbration means that a proper cofibratlon
E(X) --~e(Y)
X--~y (see w
of
induces a cofibratlon of the ends
60
The proof of Theorem (3.2.2) is contained in Propositions (3.2.6), (3.2.24), (3.2.27) and (3.2.28), below.
(3.2.6)
Proposition.
(Verification of Axiom M0).
CJ
admits finite limits
and colimits.
Proof.
Let
D
be a finite diagram in
have colimits
colim D. J
yield objects
{colim Dj}
colimit and limit
In fact, if
of D,
C
and
limits
and
CJ.
The induced diagrams
llm D. in C J
{lim Dj}
in CJ
respectively.
by Axiom M0 for
Dj over C C.
These
which are easily seen to be the
D
admits more general colimits or limits, so does
In order to verify Axioms M2, M5 and M6
for
CJ
CJ .
we shall give explicit
descriptions of fibrations (Proposition (3.2.7)) and trivial fibrations (Proposition (3.2.17))
in CJ.
(3.2.7) for each
Our descriptions will involve diagram (3.2.3).
Proposition. j
in J
A map
p:Y --->B in CJ
the induced map
qj
is a fibration if and only if
in the diagram
limk < jYk P J ~
PJ
lJmk < jPk
Bj
-
~
i~
k < jB k
81
(Pj
is the pullback) is a fibration in
Proof.
First, let
p
C.
be a fibration in
CJ,
that is, assume that
p
rlght-lifting-property with respect to the class of trivial cofibrations in We shall show that each induced map
qj:Yj ---~Pj
property by constructing suitable "test maps"
has the CJ.
has the same right-lifting-
K--~L
in CJ
which are trivial
cofibrations.
Consider a solid-arrow commutative diagram
A
:Yj
(3.2.8) /
X
in
C
K = ~
in which and
i
L = ~
> Pj
is a trivial cofibration. in C J
Define objects
as follows:
Ii
for k < J, for k = J,
L~
otherwise;
for
k~j,
otherwise . The required bonding maps are induced by
i and id X.
trivial cofibration
Diagram (3.2.8) induces a solid-arrow
commutative
diagrsm
i' :K --~ L
in CJ.
Then there is an induced
82
K
"" '.~ Y / / 1
(3.2.9)
9
i'
g,./
/
p
f / /
L
in
CJ
(the maps
Kk--~Y k
the composite maps
X--~Pj
~B
are induced from the map ~ Yk
for
k < j
g
in diagram (3.2.9).
of diagram (3.2.9), the map (3.2.8).
Hence
qj
qj:Yj --~Pj
j ffi k
p:Y-->B
The right-lifting-property
gj:Lj = X - - > y j
CJ
with the property that the induced Consider a solid-arrow
CJ
H
Y / J
/ J
(3.2.10)
f.-"
p
J 1 9p,
,,
,
h
is a trivial cofibration.
We shall obtain the required
filler f = {fj:Xj ---~Yj}
by induction on
J .
p
is the required filler in diagram
be a map in
A
i
of
Because diagram (3.2.8) is the jth level
(see diagram (3.2.3)) are fibrations.
commutative diagram in
in which the map
and
is a fibration, as required.
Conversely, let maps
for
(see diagram (3.2.8)), the other
maps in diagram (3.2.9) are defined similarly). yields a filler
A--~yj
63
Consider a fixed index fk:~--~Yk
j.
k < j,
Suppose that for all
there exist maps
with the following properties:
fkik = H k , (3.2.11) Pkfk = h k ;
(3.2.12)
f~ o bond = bond ofk
restrictions are vacuous).
(3.2,13)
for
s < k
(if
j
has no predecessors
these
Formula (3.2.12) yields a composite map
Xj
> limk < j Yk
> limk < jBj ;
by formulas O.2/1) this map is equal to the composite map
(3.2.14)
Xj
>
Bj
~ lim k < j B k.
In fact, formulas (3.2.10) - (3.2.14) yield a solid-arrow commutative diagram
H.
]
A,
Yj /
(3.2.15)
J
f
i
i
I lqj
]//
3
/ J
~.
I
,t ~
1.
X. J
in
C.
(This uses the definition of
trivial cofibration and
gj
P. J
in diagram (3t.2.3).) i
is a fibration in
C,
Because
there!exists a filler
i. 3 f.
is a in
3
diagram (3.2.15). for
Further, diagram (3.2.15) yields formulas (3.2.11) and (3.2.12)
f.. 3 By continuing inductively, we obtain the required filler
diagram (3.2.10).
f = {fj}
in
64 (3.2.16)
P!oposition.
level, the map
Proof.
pj:u
Let
--~Bj
For a given
p:X--~B
be a fibration in
is a flbration in
j in J,
Then a t each
C.
consider the commutative-dlagram
< J Yk
"'~
(3.2.3)
CJ .
\
~ '
pj\
I
'j
I~<
II
Bj
JPk
> lim k < jB k
By introducing the indexing category K=
we see that the map Hence
l~<jp
k
{k[k < J) = J,
is a fibration in
p'j, and thus the composite map
(3.2.17)
Remarks.
Trivial fibrations in
(3,2.18) Proposition.
fibrations in
C.
(apply Theorem (3.2.4)). are fibrations in
C.
D
The above proof illustrates the usefulness of cofinite
strongly directed indexing sets: maps in
only if the induced maps
pj = p'jpj
C
CJ
CJ
may be constructed inductively.
admit a similar characterization.
A map
p:@ --~ B i n
qj:Yj --~Pj
CJ
is a trivial
(see Proposition (3.2.6))
flbratlon are trivial
i f and
65
Proof.
First, let
J,
element in
the maps
p:Y--->B
be a trivial fibration.
qj:Yj --~Pj
and
pj:Yj --~Bj
If
j
is an initial
are equal.
a flbratlon by Proposition (3.2.7) and weak equivalence by hypothesis. qj(- pj)
But
pj
is
Hence
is a trivial flbration, as required.
Now suppose that for a fixed (non-lnltlal)
the induced maps
qk
are trivial fibrations.
J
in J,
and for all
We shall show that
qj
k < J, is a trivial
fibratlon.
Consider the commutative diagram
y
qj
~- Pj
'~ l ~ k < jYk
limk< jPk l~mk< js k
Bk
in which
Pj
induced map
is a pullback. Pj --~B k
hence a weak equivalence.
If
l ~ k < JPk
were a trivial fibration, the
would also be a trivial fibration (by Axiom M4 for But then
qj
would be a weak equivalence (since
a weak equivalence by hypothesis, this follows from Axiom M5 for a fibratlon by Proposition (3.2.7), so that required.
fj
C).
But
fj
is
fj
is
would be a trivial fibration, as
It therefore suffices to show that the maps
(3.2.19)
C), and
l ~ k <JPk: llmk <jYk -'-'~llmk <jBk
are trivial fibratlons.
To do this, we consider a commutative solld-arrow diagram of the form
66
K
~"limk < jYk
/
i
(3.2.20)
/
f'~"
i l imk
< jPk
/
i
1 / /
L --
in
C
in which
i
> llmk < jB k
is a cofibration.
We may define a filler
f
in diagram
(3.2.19) by defining maps
fk :L
for
k < j
~ Yk
which make the,diagrams
K
"> limk<
j~k
Yk
(3.2.21) ,
.k L
~
I im k < j I~k
~.-Bk ,
k < j
and fk
> Yk lbon d
(3.2.22~
Ys
~
commute.
This requires a second induction; this time on
fixed
aud all
k,
A < k
k.
there exist the required fillers
predecessors, this condition is vacuous).
K
We obtain a map
; Pk
Suppose that for a fA
(if k
has no
67
(see diagram (3.2.3)) for which the solid-arrow diagram
K
.
.
.
i
.
> Yk
.
/ /
qk
/ / /
(3.2.23)
> l im< < kYs
>Pk
L
I
lim~ < kp s
llm E < kBE
Bk
commutes.
But the map
qk
is a trivial fibration by our inductive assumption.
Hence there exists a filler
fk
for the upper left corner of diagram (3.2.23).
Diagram (3.2.23) immediately implies that the required diagrams (3.2.22) commute.
(3.2.21) and
Continuing inductively yields the required maps
fk
and fille~
f = l~%
in diagram (3.2.20).
Hence, the maps (3.2.19) are trivial flbrations,
as required.
The proof of the converse is similar to the proof of the "if" part of Proposition (3.2.6) and is omitted.
We may now verify t~at
D
C
satisfies Axioms M2, M5, and M6 for a closed model
category.
(3.2.24)
Proposition.
(Verification of Axiom M2)
may be factored as
x i--~ z
P-y
Any map
f:X--~ Y
in C J
68
where
i
is a cofibratlon,
p
is a flbration, and either
i
or p
is a weak
equivalence.
Proof,
We shall factor
f
as pi
with
of the other case is similar and omitted. fj :Xj - ~ Yj
i
a weak equivalence.
To factor
f,
The proof
we~shall factor the maps
as
xj
(3.2.25)
i
zj
p
where: a)
for
k < j,
b)
the maps
c)
the induced maps
i. J
pj:Zj ---~Yj
i.3 and pj
If
J
ik
and Pk'
are trivial cofibrations;
qj:Zj --~Pj
respectively;
and
associated with the maps
(see diagram (3.2.3)) are fibrations.
Suppose for a given factored.
cover
j
and for all
has no predecessors,
form the co-,,utative diagram
k < j,
the maps
fk
have been so
this condition is vacuous.
We may then
69
X.]
> llm k <jXk
(3.2.26)
limk < jZk
Qj
i ~ ~ j Pk
j
YJ
where
Qj
-
i s a p u l l b a c k , and the map
~
gj
ls
is induced from the map
in the proof of P r o p o s i t i o n ( 3 . 2 . 1 7 ) ,
see diagram ( 3 . 2 . 2 0 ) ,
a fibration.
gj
Axiom M3 for
Hence the induced map
< jYk
l ~ k < jPk"
As
limk< jPk
is
is a fibration
(by
the map
(see diagram (3.2.26))
C).
Now, f a c t o r the map
Xj --~Qj
(in diagram (3.2.26)) as the composite
Xj (using Axiom M2 for
C).
~ Zj
q'J ~ Qj
Finally, let
pj = g j q ' j : Z j
"- Qj ~ Y j
9
Then the factorization
Xj
il
~ Zj
Pl
"u
(see diagram (3.2.25)) has the required properties.
The conclusion follows.
El
70
(3.2.27)
Proposition.
(Verification of Axiom M5).
If two of the maps in the
diagram [ X
~Y
Z are weak equivalence,
Proof.
then so is the third.
This follows from Axlom M5 for
C
since weak equivalences
in
CJ
are
defined degreewise.
(3.2.28)
a)
Proposition.
(Verification
of Axiom M6).
A map is a fibration if and only if it has the right-liftlngproperty with respect to the class of trivial cofibrations.
b)
A map is a eoflbration
if and only if it has the left-lifting-
property with respect to the class of trivial fibrations. c)
A map
f
is a weak equivalence
where
v
has the left-llftlng-property
class of fibrations and
u
if and only if
f = uv
with respect to the
has the rlght-lifting-property
with respect to the class of cofibrations.
Proof.
Part a) is contained in Definitions
(3.2.1).
The proof of Part b) is similar to that of Proposition is to associate objects
to an element
E = {E k}
and
j
A = {~}
of J in C J
(3.2.7).
and a fibration defined by setting
Y--->B
The main step in C,
the
71
= ~ Y
for
k ~ j,
Ek
L = ~
*
otherwise,
B
for
*
otherwise~
k ~ j,
Ak
L
with the evident bonding maps and also the fibration
E--~B
then proceed as in the discussion following diagram (3.2.8).
in C J.
One may
Remaining details
are omitted.
For Part c), first let
f
be a weak equivalence.
is a trivial fibration and
v
is a fibration.
Factor
f
By Axiom M5 for v
(3.2.27)) and Propositions (3.2.7) and (3.2.17),
as CJ
uv
where
u
(Proposition
is a trivial coflbration.
The
required lifting properties follow easily.
Conversely, arguments similar to the proof of Proposition (3.2.18) show that maps
u
f = uv
and v
with the given lifting properties are weak equivalences.
is a weak equivalence by Axiom M5 for
C J.
Hence
Details are omitted.
This completes the proof of Theorem (3.2.2).
w
The homotopy theory of Let
C
p r o - C.
be a closed model category which satisfies Condition N (w
shall show that
pro - C
model categories
CJ
(J
required homotopy category
We
inherits a natural closed model structure from the closed is a cofinite strongly directed set); this will yield the Ho(pro -C)
(see w
One of our main tools is the
Marde~i6 trick (Theorem (2.1.6)) which states that any inverse system is isomorphic to an inverse system indexed bY a coflnite strongly directed set.
72
(3.3.1) if
f
Definitions.
is the image in
where
J
A map pro -C
f
in pro- C
of a (level) cofibration
is a cofinite strongly directed set.
cofibrations,
is called a strong cofibration f. J
in some
Strong fibrations,
C J,
strong trivial
and strong trivial fibrations are defined similarly.
A map in
pro -C
is a cofibration
of a strong cofibration.
Fibrations,
in
pro- C
are defined similarly.
if
f = pi
where
(3.3.2)
p
Remarks.
if it is the retract in Maps
trivial cofibrations, A map
f
in pro -C
is a trivial fibratlon and
Clearly,
i
(resp.,
p r o - SS
are both cofibrations
(3.3.13),
below, shows that the converse assertions hold.
By Definitions
(3.3.1)
and trivial fibrations is a weak equivalence
is a trivial cofibration.
trivial cofibrations (resp., fibrations)
(pro -C)
trivial fibrations)
and equivalences.
the classes of cofibrations,
fibrations,
Corollary
trivial cofibra-
tions, and trivial fibrations are each closed under the formation of retracts. need to show that the class of weak equivalences Definitions
(3.3.1) yield an apparently
the composition
Definitions
larger class of weak equivalences
do not know if every weak equivalence equivalence
in some
in the categories
(3.3.1) are essentially
of two weak equivalences in
CJ
than the (Defini-
forced by the requirement
yields a weak equivalence pro- C
We
is also closed under retracts.
class of retracts of the (level) weak equivalences riots(3.2.1)).
that
(Axiom MS).
Theorem.
C J.
model category.
pro -C,
We
is a retract of a (level) weak
In this section we shall prove the following.
(3.3.3)
in
together with the above structure,
is a closed
73
(3.3.4)
Theorem.
cofihrations,
flbrations,
lim:pro -C---~C
Proof.
The constant diagram functor and weak equivalences.
preserves
C
preserves
The inverse limit functor
flbrations and trivial fibratlons.
Immediate from the Definitions
The category of towers, from
C--~pro-
tow -C,
(3.1.1) and Theorem (3.3.3).
D
inherits a natural closed model structure
pro -C.
The proof of Theorem
(3.3.3) involves the following main steps:
Verification
of Axiom M0 (Proposition
(3.3.5));
Verification
of Axlom M2 (Proposition
(3.3.8));
Verification
of Axiom M6 (Proposition
(3.3.9),
Verification
of Axiom MS:
Special cases (Propositions General case (Proposition
(3.3.5)
Proposition
(3.3.15), and (3.3.17));
(3.3.18), and (3.3.26));
(3.3.35)).
(Verification of Axiom MO).
Pro - C
admits finite
colimits and limits.
Proof. a colimit;
Let
A
be a finite diagram in
the construction of a limit for
ing identity maps if necessary,
pro - C. A
is similar and omitted.
we may assume that
the original and new diagrams will be isomorphic). reindexing diagrams of
(see Proposition C
(2.1.5))
which determines
ing the Marde~i6 construction
to
A
We shall show that
A
(Theorem (2.1.6))
By insert-
Applying the Artin-Mazur
pro - C to
has
has no loops (the colimits of
yields an inverse system
a diagram in
A
{A i}
isomorphic
{A i} to
A .
yields a diagram
of ApplyA'
74
over some level category ther
4'
and
X' = colim 4'
A
CJ
indexed by a cofinlte strongly directed set
are isomorphic diagrams over
in C J
(X'
X' = colim A
coherent family of maps from the objects of of
to an object
X', Y
the maps from
to Y
some level category K
with
~,, i ~
Artin-Mazur
Let
CK
restrictions
family of maps from 4, Y,
A
and
~"
in
of
and
in the
Because any dlagram whlch represents
T: K
~".
to Y"
A' e 4' m A
To check the
indexed by a cofinite strongly directed set
in pro -C.
4"
to X'.
As above, we may define a diagram
relndexlng process yields a d i a g r a m w h i c h
the maps from 4" m T*
( m 4')
be the diagram consisting of
in pro - C.
there is a natural coflnal functor appropriate
A
(3.2.6)).
Clearly there exists a
suppose that there exists a coherent
in p r o - C. A
in pro -C. A
fur-
Now let
is defined levelwise by Proposition
We shall now check that
universality
pro -C.
J;
Let
~ J. X"
Now let
required universal property.
4"
be the colimlt of
factor uniquely through X" e T*X' ~ X'
represents
X".
in pro -C.
by restriction,
A
and Y" d"
be the
in C K.
Then
Further, Hence
X'
has the
D
Artin and Mazur give a non-constructive
proof of the following more general
result.
(3.3.6)
Proposition.
universe such that limits.
SS
[A-M,
Propositions
is U -small.
Then
A.4.3 and A.4.4]. pro -SS
Let
U
be a
admits U - small colimits and
D
(3.3.7)
Remarks.
Suppose that
C
infinite diagram over some level category
admits arbitrary colimits. C J,
and
X
Let
be its colimit in
A
be an
CJ
(X
75
It is easy to see that in
is defined levelwise as in Proposition
(3.2.6)).
general
in p r o - C.
X
is n o t the colimit of
A
The statement of the following proposition Axiom MI; see Remarks
(3.3.8)
fibration,
or
Proof.
i
(Verification
f = pi
where
f' f
f'
By Propositions
of Axiom M2).
Any map
is a trivial cofihration and p
p'i'
in
C J.
f' >X' ---~ Y'
CJ
By Axiom M2 for where
complete the proof, let
i
X
p'
and
and p
e
We shall now begin the verification proposition
(3.3.9)
CJ i'
> Z',
(Proposition
Z'
p'
"Y'
f
as the composite
f
J
-.g
pro - C.
solid-arrow diagram
'~Y
~B
Consider
we may factor To
0
p r o - C.
(3.3.10)
X
(3.2.24)),
J.
composite mappings
~'>Y.
of Axiom M6 for
Given any commutative
A
is a
have the required properties.
is a special case of Axiom MI for
Proposition.
p
~Y,
be the respective
i'
"~X'
in pro- C
is a trivial fibration.
(2.1.5) and (2.1.6) we may factor
--
f
is a level map indexed by a cofinite strongly directed set
as a map in as
i
is a coflbration and
X
where
of
(3.3.2).
Proposition.
may be factored as
is a technical reformulation
The following
76
in which either
i
is a trivial cofibration and
cofibration and
p
is a trivial fibration, there exists a filler
Proof.
p
is a fibration, or
We shall only discuss the case in which
i
i
is a
h.
is a trivial cofibration.
The proof of the other case is similar and omitted.
Since the lifting property described in diagram (3.3.10) is preserved under the formation of retracts, we may assume that
i
is a strong trivial cofibration
indexed by a cofinite strongly directed set
J
indexed by a cofinite strongly directed set
K.
and that
p
is a strong fibration
We shall now replace diagram (3.3.10) by a suitable level diagram. merely apply the reindexing techniques of w cofinal functor
T:L--~K
We cannot
since it appears unlikely that a
maps fihrations in
~
into fihrations in
CL
(fibra-
tions are no___tdefined levelwise).
Since
J-"~K
J
is cofinite, we may inductively define an order preserving function
(j +--~k(j))
and commutative solid-arrow diagrams
*j
~(j) (3.3.11)
9
ik (J)i
h'i/// / i / / /
,
Yj iPj
/ /
Xk(j)
"~
' Bj
which represent diagram (3.3.10) (that is, there are maps of diagram (3.3.10) to diagrams (3.3.11) over diagram of diagrams
pro- C
such that if
j' > j
there is a commutative
77
(3.3. i0)
j,
level
(3.3.11),
"
j
level
(3.3.11),
We thus obtain a commutative solid-arrow diagram
{r )
>
u
/ / #
//
(3.3.12)
{ik}
{h'. }/z
{ik(j) }
/
/
/
{pj }
i
/ I
) {Xk(j)}
{xk
with the following properties: f
and g
be cofinal, hence and CJ
the composites along the top and bottom rows are
respectively, and the right-hand square is a level diagram indexed by
(that is, a diagram in
CK
~ {Bj}
CJ).
Note:
{Ak} # {Ak(j)},
are defined levelwlse,
Since, by hypothesis,
{pj}
in general the function etc. {ik(j)}
is a trivial cofibration in
is a fibration in
CJ,
{h'j}
by Axiom M6 for
{h'j} ~{Xk(j)}
~ {Yj}
[]
CJ .
CJ
in diagram (3.3.12).
is clear that the composite mapping
is the required filler in diagram (3.3.10).
need not
Further, since trivial cofihrations in
(Proposition (3.2.28)), there exists a filler
{xk}
J--->K
J
It
78
(3.3.13)
Corollary.
A map is a trivial cofibratlon
a cofibration and a weak equivalence.
if and only if it is both
A similar description holds for trivial
fibrations.
Proof. f:X--~Y
The "only if" assertions hold by definition. be both a cofibration and a weak equivalence.
(3.3.1), write fibration.
f = pi
where
We shall see that
i f
Conversely,
Using Definitions
is a trivial cofibration and is a retract of
i,
let
p
is a trivial
and hence a trivial
cofibration.
i
(3.3.14)
~z
f
p
/ .f
-
Proposition (3.3.14)
;Z
(3.3.9) yields a filler
id.
g
//// I y Y
in diagram (3.3.14).
Rewriting diagram
in the form
X
Y
where
pg = idy,
Verification
shows that
g
f
;Z
P
;Y
is a retract of
of the assertion about fibrations
This answers one of the questions
i.
is similar and omitted.
raised in Remarks
(3.3.2).
D
79
(3.3.15) a)
Proposition.
A map
p
(Verification
is a fibration
cofibrations
of Axioms M6a and M6b).
if and only if for all maps
and equivalences,
the pair
(i,p)
i
which are
has the lifting
property. b)
A map
i
is a cofibration
if and only if for all maps
are fibrations and equivalences,
the palr
p
which
(i,p)
has the lifting
(3.3.13)
(which shows
property.
Proof. a)
The "only if" part follows from Corollary that
i
is a trivial cofibration)
Conversely, a). u
let
p
is a fibration and
u, b)
(3.3.9).
be a map with the lifting property of hypothesis
Use Axiom M2 (Proposition
proof of Corollary
and Proposition
v
3.3.8)
to write
f = uv,
is a trivial cofibration.
(3.3.13),
it follows that
p
where
As in the
is a retract of
and hence a fibration.
The proof is similar to the proof of a) and is omitted.
D
Similar arguments yield the following.
(3.3.16)
Proposition.
equivalence)
A map
i
is a trivial cofibration
if and only if for all f i b r a tions
p,
the pair
(cofibration and (i,p)
has the
lifting property.
A map
p
is a trivial fibration
all cofibrations
(3.3.17) equivalence p,
i,
the pair
Proposition. if and only if
the pairs
(i,u)
and
(fibration and equivalence)
(i,p)
has the lifting property.
(Verification f = uv (v,p)
of Axiom M6c).
if and only if for D
A map
f
is a weak
where for all cofibrations
i
and fibrations
have the lifting property.
80
Proof.
By Proposition
lences is equivalent
(3.3.16),
the above characterization
to that of Definitions
We have completed the verification begin the verification relatively
of Axiom M5:
(3.3.1).
of weak equiva-
Q
of Axiom M6 for
weak equivalence
p r o - C,
and shall now
is a congruence.
This
lengthy process consists of first using the lifting properties developed
above to verify Axiom M5 under the further assumption that all maps are cofibrations or all maps are fibratlons.
Secondly, we use the faetorizations
to verify the general case of Axiom M5 for
(3.3.18) cofibratlons.
Proposition.
pro - C.
Suppose that the maps
If any two of the maps
given by A x l o m M /
f, g, and
f:X--~Y gf
and
g:Y'-~Z
are
are weak equivalences,
then so is the third map.
Proof. Proposition equivalences
Case I: the pairs
There are three cases. (3.3.9)
For Cases I we use Corollary
(3.3.13) and
to characterize maps which are both cofibrations
and weak
by their lifting properties.
Let (f,p)
f
and
and
g
be weak equivalences.
(g,p)
Then for all fibrations
have the lifting property.
composite has the same lifting property;
thus
gf
Consequently,
is a weak equivalence,
P,
their as
required.
Case II:
Let
a weak equivalence
f
gf
be weak equivalences.
by verifying
that for all fibratlons
the lifting property.
and
Consider a commutative
We shall show that p,
the pair
(g,p)
solid~arrow diagram of the form
g has
is
81
fJ . X
(3.3.19)
h
y
/ / / / / / / /
/
in which
p
is a fibration.
We shall show that
required, by constructing a filler
Because the composite map K':Z -'~E
such that
We shall deform
Because category
K'
f
C J,
implies that
k
gf
K'gf = hf
K,
g
is a trivial coflbration, as
above.
is a trivial cofibration, and
pK' = k.
into the required filler
Caution:
f'
J
Hence
f':X' --~Y'
induces a trivial cofibration
f
x [0,i]
induces a trivial cofibratlon
i:Y x 0 u X x [0,1] U Y x 1 --~ u x [0,1]
in
pro -C.
K'g # h.
in some level
is a cofinlte strongly directed set, Proposition (2.3.5)
i':Y' • 0 u X' x [0,i] u Y' x i - - ~ u
in 2.
in general
K:Z---~E.
is a retract of a trivial coflbration where
there is a map
Form the solld-arrow co~,utatlve diagram
82
y x 0 u x •
[0,i]
K'g u h K a proj u h: E
u Y x 1
I"/
/' /
k ~ o proj
Y • 10,11
By Axiom M6 (see Proposition
As above, Proposition
(3.3.99,
(3.2.29)
PB
there exists a filler
K (2)
implies that the cofibration
above.
g:Y--~Z
induces
a trivial cofibration
i':z x 0 u Y x [0,i]
Now form the commutative
3
K' u K (2)
/
~ K(
/
/
K
/ /
/
z ~ [0,1]
Again, as above, there exists a filler
f
/
i'
let
x [0,I].
solid-arrow diagram
Z x 0 u Y • [0,i]
Finally,
~z
K o ~roj
>
K (3) .
be the composite mapping
K(3) Z a Z x 1
Then
K
>Z
is the required filler in Diagram
equivalence,
as required.
x I
(3.3.19)
~E.
(easy check),
so
g
is an
83
Case III:
Let
g
and
gf
be weak equivalences.
Assume,
for now, the
following l~mm~.
(3.3.20)
Lemma.
Suppose that a map
i
in pro -C
has the left-lifting-
property with respect to all fibrations of fibrant objects.
Then
i
is a trivial
cofibration.
Consider a solid-arrow commutative
diagram
~E
X
f. I i I Ikgll/lil/f / !l
P
d (3.3.21)
Y
~
i
g
/
'B
1
k/ ]il i I i I z /
1
i
iI q
r
Z
in which
p
is a fibration,
Because
g
and gf
are weak equivalences
we may successively
fillers
h
aund kg
above.
is a filler in the top square of
Diagram
(3.3.21) above,
cofibratlon
which the pair
B
is fibrant
Because
Hence,
Proof of Lemma 3.3.20. (i,p)
has the following
kg
(that is,
q
is a fibration). construct
the
has the required lifting property to be a trivial
(Lemma (3.3.20)).
(3.3.22)
a)
f
and
>*
f
Let
is a weak equivalence,
i
denote the class of all
has the lifting property.
three properties.
A pullback of a map in
i
is in
L .
as required.
p
It is easy to check that
for [
84
b)
Let
E(J)
be an inverse system of objects in
indexed by a eofinlte directed set O.
J
with a least element
If all the induced maps
E(j)
are in
L ,
is in
E(O)
[ .
To show that in
i
in
Now let directed set.
j 9 J,
is in
i .
C;
we shall show that
By Theorem (3.3.4), by property
N2
i
contains all
i contains all flbrations of
of C
(see w
i
contains all
C.
p:E-'~B Let
an inverse system For
L
is a trivial coflbration,
pro - C.
fibrant objects in fibrations
9 J
then the induced map
A retract of a map in
fibrations
j
~ l i<~j { E ( k ) } , ~
liraj{ E (j) }
c)
pro - C
be a fibratlon in J* = J u {0},
{E(J) Ij 9 J*}
define
E(j)
where over
C J,
where
0 < j pro - c
J
is a coflnite strongly
for all as follows.
j
in J. Set
Define E(O) = B.
by the pullback diagram
E(j)
~ E. 3
(3.2.23)
B
- B.
J
i~
pro-C.
That is, for
k ~ j,
E(J)k = E k,
otherwise,
E(j) k
is a
85
suitable pullback. llm {E(J)} ~ E.
Because
J
is a cofinlte strongly directed set,
We shall see that
{E(J)}
satisfies the hypothesis of
property (b) above.
For now, consider a fixed
j
in J.
Z(J)
is in
To show that the map
> i~
< j{Z(k)}
L , we shall show that given a cc~mutatlve solld-arrow dlagram of the form
h
A
Z
-'; E ( J ) ,f
p
/ / /
It/" / / /
X
in
pro -C,
in
pro - C
h2
there exists a filler
) l ~ k < jE(k)
H.
l i ~ < jE(k)
To do t h i s , form the pullback diagram
llmk < j Ek
~li~k <j{Bk}"
Thus
h2
corresponds
t o maps
h 2':x
~ l ~ k < j{Ek},
h2" :X
~B
and
86
whose images in
l ~ k < j{B k}
pullback diagram (3.2.23) in
agree.
Also, because
pro -C,
hI
E(j)
is defined by the
corresponds to the pair of maps
hI':A---~Ej,
and
hI":A---~B ,
whose images in
Bj
are equal.
Now consider the resulting commutative solid-
arrow diagram
(3.3.25)
i
/
h3 . ' ~
X
P39
Here
~B
, \/IJ
* Bj
~ I ~ k < j { B k}
is a pullback, and the composite mapping
A---~X--~B
the universal property of pullbacks, there is a unique filler (3.2.7), qj
is a fibration in
h4
Diagrams (3.2.23) and (3.2.25) show that the maps
above.
~:X--~Ej
C , hence
induce a unique map
diagram (3.2.24).
H:X---~E(j),
Therefore
L
By
By Proposition
h2":X--->B
and
which is the required filler in E(j)
satisfies the
Hence the induced map
p:E = lira {E(J)} [ .
h 3.
hl".
so that there exists a filler
We have shown that the inverse system
hypothesis of condition (b) above.
is in
qj E i ,
is
~E(O) = B
contains all strong fibrations in
pro- C.
By
87
property
(c) above,
trivial cofibration,
[ contains all flbrations as required.
in
pro-C.
Therefore
i
is a
g:Y--->Z
be fibrations.
If
are weak equivalences,
so is the third.
Q
Similar techniques yield the following.
(3.3.26)
Proposition.
any two of the maps
Let
f, g, and
The proof is analogous
f:X-->Y, gf
and
to the proof of Proposition
(3.3.18) and requires a
lemma "dual" to Lemma (3.3.20).
(3.3.27)
Lem~a.
Suppose that a map
property with respect to all cofibratlons
p
in pro -C
has the right-lifting-
of cofibrant objects.
Then
p
is a
trivial fibration.
The proof is similar to the proof cofibrations
in
CJ
Proposition and L e m a
(3.3.22), and somewhat simpler because
are defined levelwise.
Details of the proofs of the above
are omitted.
We now begin the proof of the general case of Axiom M5 with four preliminary lemmas.
(3.3.28) f = pi,
Lemma.
where
i
Let
f:X--->Y
be a weak equivalence.
is a trivial coflbratlon and
p
Suppose that
is a flbration.
Then
p
is
a trivial fibration.
Proof.
By Definitions
trivial cofibration and arrow diagram
p'
(3.3.1), we may wrlte is a trivlal fibration.
f = p'i',
where
i'
Form ~he commutative
is a solid-
88
x
/ (3.3.29)
Z
.
f
.
4-
.
.
.
.
.
.
-~ Z !
.
. . . . . . . . .
By Axiom M6a (Proposition (3.3.15)), there exist maps
f and f'
(as shown
above) such that diagram (3.3.29) together with either dotted arrow commutes. Hence
f'fl = i
and
pf'f = 0.
Thus there exists a commutative solid-arrow
diagram (see Proposition (2.3.5))
td Z u ieproj Z • 0 u Xx
[0,1]
u Z •
u f'f
1
> Z J I i el J J
p
J I I I
/,
I
~V
As in the proof of Case II of Proposition (3.3.18), Hence there exists a filler which covers
i~.
H,
that isp a homotopy
j
is a trivial coflbratlon. H:Id Z : f'f
A similar construction yields a homotopy
Therefore the flbratlons
p
and p'
relative to
H':idz, = ff'.
are flber-homotopy-equlvalent.
Hence they
have similar lifting properties (use deformations analogous to those in the proof of Case II of Proposition (3.3.18)), so that Proposition (3.3.16).
0
p
X
is a trivial flbratlon by
89
(3.3.30) f = pi,
Lemma.
where
p
Let
f:X--->y
be a weak equivalence.
is a trivial fibration and
i
Suppose that
is a cofibration.
Then
i
is
a trivial cofibration.
The proof is similar to that of Lemma (3.3.28), and is omitted.
At this point we recall Condition N3 for
C:
at least one of the following
statements holds.
N3a.
All objects of
C
are cofibrant.
N3b.
All objects of
C
are fibrant.
We shall assume N3a for the remainder of this section, unless otherwise specified.
If instead N3b holds, replace " f i b r a t l o n " b y
(3.3.31), below, obvious changes.
(3.3.31) a section
"dualize"
"coflbration"
in Lemma
the proof of Lemma (3.3.32), below, and make some other
Details are omitted.
Le-~a. s:B ---~E
Let
p:E--~B
(that is,
be a trivial fibration.
ps = idB) ;
Then there exists
further, any section is a trivial
cofibration.
Proof. (3.3.9))
By Assumption N3,
yields a filler
s
B
is cofibrant,
in the commutative
so Axiom M6 (see Proposition solid-arrow diagram
/ / /
I Jl /
/
/
/
B
'
~B
90
Now, let
s':B---> E
be a section to
there exists a homotopy
H
(3.3.32)
Let
a trivial cofibration.
Proof. i
over
id B
from
s'
to
id E,
As in
f:X--~y
be a trivial fibration and
Then the Composite
p
w
is a trivial cofibration.
gf
g:Y--~Z
be
is a weak equivalence.
By Axiom M2 (Proposition (3.3.8)) we may factor
is a cofibration and
sp
(3.3.18), Case II, or the proof of [Q -i, Lemma 1.5.1,
to show that
Lemma.
As in the proof of Lemma (3.3.28),
H:E • [0,I] --~E
the proof of Proposition we may use
p.
is a trivial fibratlon.
gf
as
pi,
where
Form the commutative solid-
arrow diagram
i
x
(3.3.33)
f
Ii
--~ w
],,:
s
Y
Lemma (3.3.31) yields a section
s
t
$
~ Z
to f .
Because
pis = gfs = g,
there
exists a co~nutative solid-arrow diagram
y
is
9
~.
/
/
/ /
t
(3.3.34)
/
/ / /
Z
Axiom M6 (see Proposition map
t
is a section to
id
9
(3.3.9)) yields a filler p
and satisfies
tg = is
t
in diagram (3.3.34). (see diagram (3.3.33)).
The
91
Axiom M5 for cofibrations
(Proposition
are trivial cofibratlons. (3.3.1)), as required.
(3.3.35)
Hence
(3.3.18))
gf
(= pi)
is a weak equivalence
i
and
(Definitions
D
Proposition.
(Verification
X
be a ~ m ' a m
tg(= is)
implies that
in which two of the maps
of Axiom MS).
f~ Y
~
Let
Z
f, g, and gf
Then
are weak equivalences.
so is the third map.
Proof.
There are three cases.
Case I: write i
Let
f = pi
and j
f
and
and g g = pj,
write
is a trivial cofibration. composite mapping
qr
Case II.
q
where
Let p
and
k
as
(qr) (ki),
are trivial fibrations (3,3.32)
where
By Propositions
f
gf
r
and gf
is a trivial cofibration. where
ki
and
the composite mapping
jp
is a trivial fihration and
is a weak equivalence,
be weak equivalences.
is a trivial flhration, Then
(3,3.1),
k
(3,3.26) and (3.3.18) respectively,
( = qrki)
gf = qJpi.
We may therefore write
is a fibration
and q
By L ~ a
jp = rk,
Hence
is a fibration.
equivalence.
qr
p
As in Definitions
is a trivial fihration and the composite mapping
trivial cofibration.
and
where
are trivial cofibrations.
is a weak equivalence,
g = qj,
be weak equivalences.
i
and j
Write
where
is a
as required.
f = pi
and
are trivial cofibrations,
By Lemma (3.3.32),
Jp = rk,
ki
r
jp
is a weak
is a trivial fibration
We have thus factored the weak equivalence
is a trivial flotation
(by Axiom M6a (Proposition
the
(use Proposition
~3,3.18))
gf and
(3.3.15)) which implies that the class
92
of fibrations is closed under composition). fibratlon. so that
Finally, Proposition (3.3.26) implies that g(=qj)
Case Ill. g
and f
By Lemma (3.3.28),
is a trivial
is a trivial fibration,
is a weak equivaleuce as required.
Let
so that
q
qr
g
and gf
be weak equivalences.
gf = qjpi,
are trivial fibrations, and
i
Lemmas (3.3.32) and (3.3.30).
where
j
is a trivial coflbratlon,
is a coflbration.
pro - C
p
and q
Proceed as in Case If, using
Details are omitted.
This completes the proof that
As in Case II, factor
D
is a closed model category.
We Shall conclude this section by describing cofibrations and trivial cofibratlons up to isomorphism.
This extends J. Grossman's [Gros -i, w
tion of cofibratlon8 in his closed model structure on
(3.3.36)
Proposition.
A map
trivial cofibration) if and only if
The " o n l y
i f tl
cow -5S.
f
in pro -C
f
is isomorphic to a strong coflbratlon
(respectively, strong trivlal coflbratlon)
Proof.
is a cofibration (respectively,
(in some
part is obvious.
CS).
For t h e ttifft p a r t ,
i f n e c e s s a r y ( P r o p o s i t i o n ( 3 . 1 . 4 ) and Theorem ( 2 . 1 . 6 ) ) f ffi {fj}:{Xj} ---~{Yj}, {fj} tion)
j ~ J,
characteriza-
with
J
cofinite.
first
reindex
f
so t h a t By Definitions (3,3.1),
is a retract of a strong cofibration (respectively, strong trivial cofibra{f'k}:{X' k}
c~utative
diagram in
~ {Y'k }, pro -C
k E K,
with
K
cofinite.
Form the following
93
{xj I
id
/ {x'
{fj)
{fj}
(3.3.37)
f'k } id
{Yj } - -
(y.}
3
\ (Y k }
Z
We shall use diagram (3.3.37) to construct a level cofibration level trivial cofibration)
f"
square of diagram (3.3.37).
(3.3.38)
if
isomorphic to
f.
(respectively,
First consider the left front
We shall say that
(f'~:X'~
~Y'~)
<
(fm:Xm ~ )
t h e square (shown i n p e r s p e c t i v e )
X
m
x'
f
(3.3.39) Y
fl
s
m
yl
is a left front square of diagram (3.3.37).
Similarly, use the right front square
94
of diagram (3.3.37) to define relations of the form
(3.3.40)
(fm:Xm
>Ym ) < (f'n:X'n
>Y'n ).
By diagram (3.3.37), relations (3.3.38) and (3.3.40) and their composites yield an inverse system of maps
~
~
= {fk:Xk ~ Y k
indexed by a (cofinite)
Further,
},
strongly directed set
{f'j:X'j
~ Y'j },
J ~ I,
k E K
K.
to~ether with bondin~ maps induced by
diagrams (3.3.37) - (3.3.40) is a coflbratlon (resp., trivial cofibration) and is cofinal in
f,
hence isomorphic to
subsystem which is cofinal in (3.3.40)).
f
Remarks.
defined levelwise.
Also,
{fi:Xi
~ Yi}
admits a coflnal
(the bonding maps agree by diagrams (3.3.37) -
The conclusion follows.
(3.3.41)
f.
D
The above proof used the fact that cofibratlons in
CJ
are
We do not know whether the analogue of Proposition (3.3.36) for
fibratlons holds.
w
Suspension and loop functors, cofibration and fibration sequences. D. Quillen [Q -i, w167
-3] developed a general theory of suspension
functors and cofibration and fibratlon sequences in model category. structures on
Ho(C),
where
C
and loop is a closed
We shall sketch this theory within the context of our closed model pro-C.
95
(3.4.1)
Morphisms i__nn Ho(pro -C).
a fibrant object in
pro - C.
Let
X
[X,Y]
pro - C)
be
= IX,Y],
denotes the set of homotopy classes of maps from
with respect to the cylinder
description of
Y
Then
Ho(pro-C)(X,Y)
where
be a cofibrant object and
Ho(pro - C)(X,Y).
X ~
[0,i].
X
to Y
(in
There is another dual
Factoring the diagonal map
Y --> y x y
as
the composite of a trivial cofibratlon followed by a flbration
y
y[O,l].
yields the cocyllnder functorlally upon maps
Y
f,g:X ~ Y
P0
Now, let
in general
y[O,l]
H
; y~O,l,[]
~
y
>y
x
need not depend
One can easily show that
~
g
y
denote the projections onto the first
y x y~
C,
Caution:
are homotopic if and only if there exists a commutative diagram
and Pl
factors in
>y • y
(compare with Definition (2.3.4)).
X
here
~y[0,1]
(yO)
and second
(yl)
respectively.
be a pointed closed model category, that is, a closed model
category which is also a pointed category.
Then
closed model category (the point
is also the point of
* of C,
shall follow the "usual" conventions and write
v
pro - C,
becomes a pointed pro -C,).
for the sum (coproduct) in
We
96
p r o - C,.
(3.4.2)
Definitions.
shall write
* vx Y
If
f:X--->Y
for the cofibre of
is a cofibration in f
>,
Y
f:X--->Y
fibre of
f
>*
is a fibratlon in
pro-C,,
.
we shall write
* Xy X
for the
.xyX
>X
,
>Y
(level) coflbration
is just the levelwise
similarly the fibre of a strong (level) fibration is the levelwise fibre.
(3.4.3)
Suspensions and loop spaces.
Choose a cylinder object i 0 + if: X v X
~X
u
X x [0,I],
v [0,i].
spaces are defined dually. object
VxY
defined by the pullback diagram
Note that the cofibre of a strong cofibre;
we
defined by the pushout diagram
X
If
pro - C,
and let
Let
and let
We shall call Let
~u
Y c pro - C,
X ~ pro -C. EX EX
be c o f i b r a n t .
be t h e c o f i b r e a suspension of
be fibrant.
be the fibre of the map
o f t h e map X.
Loop
Choose a cocyllnder
(p0,PI):Y [0'I]
>y x y .
97
We shall call
~Y
is flbrant.
a loop-space of
Caution:
(see Definition
Y.
In general
Note that
E
(2.3.4) and paragraph
and
~
EX
above is cofibrant and
need no__~tbe functors on
pro -C,
(3.4.1)).
On the other hand, Quillen proves the following
theorem for arbitrary closed
pointed model categories.
(3.4.4)
Theorem
[Q-I,
pair of functors on all of
w167
X
is cofibrant and
and
~
to an adjolnt
Ho(pro -C,):
Ho(pro -C,)(ZX,Y)
If
E
We may extend
Y
ffi Ho(pro-C,)(X,nY).
is fibrant,
[EX,Y] ~ [X,~Y].
E
and
are defined up to canonical
Ho(pro-C,)(E-,-) (pro -C,) ~
isomorphism.
= Ho(pro-C,)(-,n-)
• (pro -C,)
Also,
as functors from
to groups.
The proof is similar to the proof for the category of pointed spaces, except that somewhat more care is needed because of the choices involved in defining E
and
~.
structure
The group structure on EX
> ~X v EX
the corresponding
H -structure
(3.4.5) Top,.
Let
Ho(pro -C,)(EX,Y)
which makes
group structure on
~y x ~y--->~y
Short cofibration
EX
comes from a
a cogroup object in
Ho(pro -C,)(X,~Y)
which makes
gq
sequences.
Let
Ho(pro
-C,);
comes from an
a group object in
f:A--->X
co-H
Ho(pro -C,).
be a cofibration
in
~Y
98
Mf = A • [0,i]
be the mapping cylinder of
f,
UAX/* x [0,i]
and consider the induced cofibration
i0 A
If
Ci0
denotes the cofibre of
reduced suspension of
A,
>Mf
i0,
CA
the reduced cone on
A,
and
ZA
the
we may form the sequence i0
A
> Mf
> C. 10
and induced diagram
Ci0 <------- CA u A Mf
Further, in
Ho(Top,)
of the cogroup object
~ EAv
the composite mapping
(co -H
needed.
If
A--->X
* --->X is a map in
ZA
is a coaction
on C. 10
to arbitrary cofibrations
the full subcategory of coflbra~t objects in tion means that the map
n:C. ---->ZA v C. 10 10
space with a co-inverse)
We shall extend the above observations
Ci0
pro - C,.
is a cofibratlon.) pro- C,
in
(pro- C,~,
(The "cofibrant" condiThe following notation is
we shall write
A
Q
[0,i] v A X
99
for the cofibre sum (pushout)
A
[O,l]
A |
More generally, il:A
[0,1],
i0:A---->A ~
[0,i].
in
pro -C..
cogroup
A
Let on Z .
-v A A ~
and
A--*X
Now let
A ~
we shall write
>A ~
>A
[0,i] [0,i]
be a cofibration Z
[ 0 , 1 ] vA X .
|
v A-
to denote the cofibre sum with
in
he the cofibre of
(pro -C.) c, A.
(3.4.6)
x |
vA X
,*
[0,i]
[0,I] vA X
>X
A ~
A
is cofibrant
diagram
v A A (~) [ 0 , i ]
, z vx x |
X
in which the cylinder object
that is,
We shall define a coaction of the
To do this, form the commutative
A (~ [ 0 , I ]
X vA A ~
to denote the cofibre sum with
vA X
[0,1]
~Z
[0,I]
is obtained by factoring
as a coflbratlon
the natural map
followed by a trivial fibration
100
X vA A ~
[0,i] vA X
)X ~
[0,i]
)X
(see Proposition (2.3.5)).
By construction both squares in diagram (3.4.6) are
pushout squares.
i
Further,
(see Proposition (2.3.5)) and
coflbratlons, hence so are their respective pushouts (3.4.6) induces the composite mapping
i'
> ZA v Z
c:Z
i0
are trivial
and i0'.
Diagram
Ho(pro- C,)
in
defined
below.
z
[iO]>z
vx x |
[0,1]
) ~ v A A (~) [ 0 , 1 ]
vA X
) * vA A ~) [0,i] vA *
-" E A r
It is easy to check that, in
A
on Z
and that
c
Ho(pro -C,),
c
Ho(pro - C , )
A'
which for some cofibration
constructed above.
is a coaction of the cogroup
is independent of any choices made above (see [Q- i,
coflbratlon sequence in
(3.4.8)
Z
Z
Proposition 1.3.1 and the followlng remarks]).
(3~4.7)
v
) X'
A--*X
A
>X
We therefore define a short
to be a diagram in
9 Z',
in
Z'
) ZA' v Z',
(pro -C,) c
~ Z,
Z
Ho(pro - C,)
is
9 ZA v Z
isomorphic
to the diagram
101
(3.4.9)
Proposition.
induces a short cofibration
A short cofibratlon
sequence
A
~ZAvZ
Z
~Z,
> Z
>ZA
> ZA,
where
ZA
> ZX v ZA,
is the composite
Z
and the coaction
Z
sequence
X
the "connecting map"
>X
> ZA v Z
> ZA,
ZA
> ZX v ZA
is the composite
ZA
> ZA v ZA
- i d v i d > ZA v ZA
-id:ZA--~ZA
is the inverse in the cogroup
> ZX v ZA,
ZA.
The proof is similar to the usual proof of the corresponding Ho(Top,),
and dual to the proof of [Q -1~ P r o p o s i t i o n
1.3.3].
assertion
in
Details are
omitted.
(3.4.10) A--->X-->Z~ (Barratt-Puppe
Lon~ c o f i b r a t i o n Z
>ZA v Z
in
sequences. Ho(pro -C,)
A short cofibration
sequence
induces a lon$ cofibration
sequence)
A f ---> X ---> Z ---> ZA - - ~ 9 9 9 --~ ZnA - - > z n x ---~ znz ---> zn+IA --~ " 9 9 .
Also,
for any object
Y
in p r o - C , ,
the sequence
sequence
~02
(3.4.11)
> Ho(pro -C,)(Enz.Y)
> Ho(pro -C,) (Enx.y)
> Ho(pro -C,)(EnA.y)
)
.~
) Ho(pro -C.) (Z.Y)
> Ho(pro -C,)(EA.y)
> Ho(pro -C,)(X.Y)
; Ho(pro -C.) (A.Y)
has the usual exactness properties:
a)
Sequence (3.4.11) is exact as a sequence of pointed sets, and maps of pointed sets;
b)
Sequence (3.4.11) is exact to the left of
Ho(pro-C,)(EX,Y)
as a sequence of groups and homomorphlsms; c)
Two maps in
Ho(pro - C,)(X,Y)
Ho(pro -C,)(X,Y) the group
d)
Two maps
Ho(pro -C,)(Z.Y) map
h
if and only if they differ by the action of
Ho(pro-C,)(ZA,Y)
gl,g 2
have the same image in
on
Ho(pro -C,)(Z,Y).
Ho(pro - C , ) ( A , Y ) if and only if
have the same image in g2 = ( Z f ) h o gl
for some
in Ho(pro-C,)(EX.Y).
As above, compare the usual exactness properties of Barratt-Puppe sequences and [Q-I, Proposition 1.3.4'] for the proof.
We now summarize the properties of coflbratlon sequences.
(3.4.12)
Proposition.
(dual to [Q-I, Proposition 1.3.5]).
short coflbration sequences in
Ho(pro,-C,)
The class of
has the following properties:
103
a)
Any map
f:X--->Y
in
Ho(pro -C,)
may be embedded in a cofibratlon
sequence
f
X
b)
>Y
~Z,
Z
9 ZX v Z.
Given a commutative solid-arrow diagram
X
>Y
>Z
Z I I
f
(3.4.13)
g
[
h
b
>ZXvZ
i i i i h) ) ) )
X' -->Y' -->Z'
f v h
Z' -->ZX' V Z'
in which the rows are short cofibration sequences, the filler
c)
If the maps
f
is the filler
d)
and g
h
exists.
in diagram (3.4.13) are weak equivalences, so
h.
Proposition (3.4.9) holds.
We omit the proof.
The following straight-forward proposition yields many cofibration sequences in
Ho(pro -C,).
(3.4.13)
Proposition.
Let
cofibratlons of cofibrant objects
{Aj - - ' * 5 -->Zj) Aj --->Xj,
indexed by a cofinite strongly directed set
be an inverse system of
with cofibres J.
Z.,3 over
Then tahere is an induced
cofibration sequence
r
C,
---> (xj} ---> {zj~}, (zj} =-)zr
v (z.}j
104
in
Ho(pro -C,).
S
All of the above theory may be dualized to obtain short and long fibration sequences.
Quillen [Q - i, w
discusses fibrations explicitly.
We shall
summarize this discussion below.
A fibration
p:Y--->B
in
(pro -C,)f
induces a short fibration seqUence in
(3.4.14)
in which
m
Ho(pro -C,).
F
%
n
Pro - C,)
Ho(pro -C,),
P~B,
fib • F m--~->F,
RB
on F
in
There is also an induced Short fibration sequence
RB
~F
~ E,
RE • RB
n > RB,
is the composite map
id • *
RB
and
is fibrant in
B
is a well-defined action of the group object
(3.4.15)
where
~ Y
(that is,
>RB
•
m
F
>F,
is the composite map
RE x RB
> RB x ~B
-id • id
~ RB x RB
> ~B.
This is [Q- i, Proposition 1.3.3], compare Proposition (3.4.9).
Hence, there is
an induced lon~ fibration sequence
(3.4.16)
"'-
>Rn+IB
~n%
>
B
,RnE
~F
~F
,,RnB
9 E -------~
B
> "'"
105
with exactness properties analogous to properties (a) - (d) of long cofibratlon sequences (see Paragraph (3.4.10) above)[Q-l,
Proposition 1.3.4].
We shall need the following d u a l of Proposition (3.4.13).
(3.4.17)
Proposition.
Let
fibratlons of fibrant objects
~ Ej
{Fj Ej
and Bj,
indexed by a cofinite strongly directed set
~ Bj}
be an inverse system of
with fibres J.
Fj
over
* ~ Bj,
Then there is an induced fibra-
tion sequence {F.}3
in
~ {E.}3
9 {Bj},
> {F.}j
Ho(pro -C,).
Proof.
We first replace
replace the map
{Ej}
{B.}3 by a fibrant object by a flbration
{B.}j
(Recall that the natural functor structures.)
Caution:
C, J
recall that fibrations in
is Just the levelwise fibre. first to factor the map
{E'j}
> pro - C,
but the fibre of a strong fibration in
C, J)
~{Sj} x {Fj}
pro- C,,
{B'j},
~ {B'.}] in C, J.
preserves the closed model C, J
are not defined levelwlse,
that is, a fibration in
{Bj} ---~*
as a level trivial cofibration followed
(ij) {Bj} B'.J
fibrant)
C, J ,
Use the proof of Proposition (3.2.24) (Axiom M2 for
by a fibration
( t h i s makes
and n e x t
>,
; {B'S )
and then to f a c t o r t h e composite map
{Sj}
> {Bj}
~ {B'j}
106
as a level trivial cofibration followed by a flbration {i'j} {Ej}
Let
{F'j}
{p'j) > {E'j}
be the (levelwlse) fibre of
tive diagram in
~ {B'j}.
We obtain the following commuta-
{Pj}.
J C, .
(F_)
, (E.) ---~
]
{Bj}
3
(3.4.18)
{i' .} J
}
{p l .}
{F' }5
In diagram (3.4.18)
{i"j}
>{E'j}
-] {B'.}]
is the restriction of
Regard diagram
{iwj}.
(3.4.18) as an inverse system made up of the diagrams
Fj -
> Ej
>B
(3.4.19)
i.
3
F'j
in
C,.
Because
~E'j
pj and p'j
>B'j
are fibrations (a fibration in
C,
J
is a level-
wise fibration, see Proposition (3.2.16), although the converse assertion need not hold), and lences.
ij
and i'j
are weak equivalences, the maps
Hence the diagrams
i"j
are weak equiva-
107
are isomorphic over
(3.4.20)
)
{F'j}
> {E'j}
Ho(pro -C,).
Remarks.
tion sequences in
{Fk}
Diagrams
{Ej}
>
{Bj} ,
> {B'j}
The conclusion follows,
l]
(3.4.19) may be extended to maps of short fibra-
Ho(C,).
F. .)
)E. J
)B. J
~B
E lj
>E'j
> B'j,
J
x F
>F.
J
J
(3.4.21) ~B'j x F'j - - > F ' j
However diagrams (3.4.21) do not yield the required action map
~{B'.} • {F'.} J
in
Ho(pro -C,)
(3.4.22)
(not
Remarks.
pro -Ho(C,)),
w
~ {F'j}
without suitable coherency conditions.
As in Quillen's discussion
Toda brackets and similar constructions applications
J
in
so far of these constructions,
[Q -i, w
Ho(pro - C,).
we may obtain Because we have no
we omit the details.
Simplieial Model Structures. In this section we shall prove that a simpliclal closed model structure (satis-
lying condition N of w
on C
induces such a structure on
results can be readily extended to pointed categories;
pro -C.
These
details are similar and
108
omitted.
For a finite simplicial set
K
and X, Y,
in C,
let
X~K
denote the "tensor product" and "function space" constructions in HOM (X,Y) connecting
(3.5.1)
is a simplicial set), and let SS
and
C;
see w
Definition.
H0M (K,X)
Let
and C
HOM (X,Y)
(recall that
denote the "function space"
{Xj}, {Yk} ~ C.
Let
( 5 } ~ K ffi {Xj ~) K},
HOM (K,{Xj})
and
= {HOM (K,Xj)}
together with the induced bonding maps, and let
HOM ((Xj}, {Yk}) ffi llm k collmj (HOM (Xj,Yk)}.
These constructions extend to functors HOM:
(finite simpllcial sets) ~
HOM:
(pro - C) ~
• pro- C -
• pro - C
) SS,
~
:pro -C • SS 9 ) pro- C,
* pro - C,
respectively.
and
Axiom SM0, and the following
propositions are immediate consequences.
(3.5.2)
Proposition.
For
{X.}
-
0 -slmplices of
and
{Yk } in pro -C,
3
HOM ({Xj}, {Yk}),
R0M ({Xj}, {Yk})0 ! pro =c((xj}, {Yk)), naturally in
{Xj}
and
{Yk }.
the set of
109
(3.5.3) let
Theorem
(Exponentlal Law).
{X.}3' (Yk } E p r o - C .
Let
K
be a finite simpliclal set and
Then
HOM ({Xj} (~ K, (Yk)) ~ HOM (K, HOH ({Xj), {Yk})), naturally in all variables. struction in both
Proof.
pro- C
(HOM and
is used to denote the "function space" con-
SS.)
Because
HOM ((Xj} |
K, {Yk}) ffi HOM ((Xj |
K}, {Yk})
ffi limk(colfmj(,O~ (Xj | ffi lim k {HOM ({Xj) |
and
H0M ( K , ? ) : S S - - ~ S S
c a s e where
{Yk}
preserves limits,
is an object
Y of
K. Yk)))
K. Yk)}.
we may r e d u c e t h e g e n e r a l c a s e t o t h e
C.
In this case,
H0H ({Xj) | K,Y) = colimj{HOH (Xj |
~,Y))
= col~jTHOM (K, RO~ (XS,~))}.
Because
category
{j)
K
has finitely
many n o n - d e g e n e r a t e s i m p l i c e s ,
and t h e i n d e x i n g
is filtering,
colimj~HOM (K, HO~ (Xj,Y)))
ffiHOM (K, dolimj~HOM (Xj,Y)})
ffiHOM (K, HOM ({Xj), Y)),
as required.
Naturallty follows easily.
0
110
(3.5.4)
Corollary.
With
K,{Xj},
and
{Yk }
as above,
pro- C({Xj} ~ K,{Yk}) = SS(K, HOM ({Xj},{Yk})),
naturally in all variables.
(3.5.5)
a)
D
Remarks.
The corresponding assertion for is proven similarly.
b)
examples, use the fact that
MAP (K,-): Top---~Top
(3.5.6)
Theorem
cofibration in
K; to construct counter-
H0M (K,-):SS -->SS
commute with collmits for infinite
for non-compact
(Definition (2.4.2))
Details are omitted.
The above results fail for infinite
space
(K,X)
HOM
K.
does not
Similarly the function
does not commute with expanding colimits
K.
(Verification of Axiom SM7).
pro -C,
and let
p:{Ys
--->{Bm}
Let
i:{Aj} ----->{Xk}
be a fibration in
Then: a)
The induced map
(3.5.5)
HOM ({~},{YA})
is a fibration in
b)
If either
SS
i
q ,HOM ({Aj},{Yg})
(i.e., a Kan fibratlon);
or p
is also a weak equivalence, then the map
above is also a weak equivalence.
be a pro - C .
111 Proof. a)
Consider a solid-arrow commutative diagram in
SS
of the form
HOM
({Xk},{Bm})
(3.5.6) Vn'k
An
in which k th
face.
(3.5.7)
V n'k
f
g
>
HOM ({Kk},{Ys
((Aj},{Ys
;HOM
is obtained from
The maps
g
and q
x
H0M
DA n ,
({Aj},{Bm})
the boundary of
An ,
by deleting the
correspond respectively to pairs of
g,:^n
, Ho~ ((Aj),(Y~)),
g":A n
> HOM ({Xk),{Bm});
({Aj},{Ys
q':HOM ({Xk},{Ys
> HOM
q":HOM ({Xk},{YA})
, HOM (~xk),(Bm));
such that the appropriate composite maps into
HOM ({Aj},{Bm})
Applying the exponential law (Theorem (3.5.3)) to
maps
are equal.
f, g', g", q', and q",
and
assembling the induced maps with the above coherence data (Diagram (3.5.6)), we obtain a commutative solid-arrow diagram
112
(3.5.8)
> (Ys
{Aj} ~ An u {Aj} ~ Vn'k{~} ~ Vn'k
(xk} |
"(B }.
An
m
i:{Aj} "--*{X k}
By Definitions (3.3.1), the map coflbratlon
is a retract of a levelwlse
By Axlom SM7 for
{i'r}:{A' r } --->{X'r}-
C,
(see w
the induced
maps
(3.5.9)
A'
r
~
X'
An U A'
~
vn,k
~
V n'k
> X'
r
~
An
r
r are trivial coflbrations; hence the induced map
(3.5.10) (i'r},:{A'r}~)An
u
.{X' } ~) Vn'k {A,r } Q
is a trivial coflbration.
Because
vn,S
i,
r
is a retract of
i,
is also a trivial cofibratlon; hence Axiom Ml for
h'
in diagram (3.5.8).
yields a map the map required.
h
{X'r}| {i'}, pro - C
by construction, yields the filler
Applying the exponential law (Theorem 3.5.3)) to
h:An---->HOM ({~},{Y~}). makes diagram (3.5.6) commute.
h'
By construction ((3.5.7) - (3.5.10)), Thus
q
is a (Kan) flbration, as
b)
If the map
i
is a trivial cofibration, then
i
induces a
trivial cofibration
(3.5.11)
i#:{Aj} @
An U {Aj} •
by analogues of (3.5.9) with
V n'k
~An { ~ } @
replaced by
DAn
DAn;
analogue of diagram (3.5.6) with the same replacement.
' {~}
@ An
hence a filler in the Thus
q
is a trivial
(Kan) flbratlon, as required.
Finally, suppose that (3.5.11) induced by
i
p
is a trivial flbratlon.
The coflbratlon
has the left-llftlng-property with respect to
i# p,
so the
required fillers in suitable analogues of diagrams (3.5.8) and (3.5.6) (see above) exist.
Thus
q
is a trivial (Kan) flbratlon, as required.
Theorems ( 3 . 5 . 3 )
and ( 3 . 5 . 6 ) ,
a c l o s e d model c a t e g o r y
(w167
together
- 3.3),
with the earlier
imply that
D
proof that
pro - C
pro - C
is a simplicial
model c a t e g o r y .
w
Pairs. We shall use the Bousfield-Kan [B-K] model structure on
More precisely,
Maps(pro -C)
is the category whose objects are maps
A in
pro-C,
Maps(pro-C).
~X
and whose morphlsms are commutative squares.
A map
is
closed
114
(3.6.1)
A
1
X
in
Maps(pro- C)
a fibration if
g
are fibrations,
lifting property is satisfied.
-~Y
g
is a weak equivalence and f
>B
f
if
g
and f
are weak equivalences,
and a cofibration if the appropriate
Explicitly,
(g,f)
is a coflbration if
f
and
the induced map
XJJ_ B A
are cofibrations in
pro -C
(X~
B
>Y
is the pushout obtained from most of
A diagram (3.6.1)).
(3.6.2)
Definition.
Maps(pro -C)
Let
(C,
consisting of maps
pro - C) A-->X
be the full subcategory of with
X
in
C .
It is easy to prove (as in w
(3.6.3)
Theorem.
The closed model structure on
closed model structures on
Maps(pro - C),
It is conven/ent to represent
x-
(%<
%(
---];
and a level map
tow- C)
f0:X0 ----~Yo
consists of a cofinal subtower {Xm } n
pro -C),
as follows.
a morphism~onslsts of amap
together with a compatible map f:{X m) ---> {Yn }
(C,
(C,
>{Yn }.
pro - C
in C.
induces natural and
(C,
Objects are towers
f:X-~Y
in tow-C
Alternatively,
{Xmn } = {xm}
tow -C).
with
a map
x m0
ffi X 0
115
3.7.
Geometric Models.
We shall discuss geometric models of
Ho(Top,
tow -Top)
using filtered spaces and a telescope construction. Ho(Top,
tow -Top)
Ho(tow -Top)
will be used in proper homotopy
and
Ho(tow - Top)
The geometric model of theory in w
The model of
was used in shape theory by the f~rst author and R. Geoghegan
(unpublished).
(3.7.1)
Definitions.
A filtered space
together with a sequence of closed suhspaces Xn c int Xn+ I.
A filtered map
map such that for each number A filtered space
X
consists of an underlying X = X 0 = X 1 D X 2 ~ "'',
(of filtered spaces)
f:X-->Y n ~0
X
there is a number
m ~ 0
is a continuous
Filt
and its associated
under the relation of filtered homotopy)
(3.7.2)
Definition.
is the space
Tel (X) = X 0 x 0 o bond Xl x [0,i] u bond X2 x [1,2] u bond
shown below.
1
(quotient
Ho(Fi~t).
X = {X n}
2
c Yn"
There results
filtered homotopy category
The telescope of a tower
0
f(X)
X • [0,i];
this yields a natural notion of filtered homotopM of filtered maps. a filtered category
X
with each
with
induces a natural filtration on its cylinder
space
"'"
116
Tel (X)
is filtered by setting
Tel (X) n 9, X n x n u
bond Xn+l x [n, n + l ]
u bond "''.
This construction extends to a functor
Tel:
Further,
Tel
(3.7.3) of
Filt
takes a
Top N
"x [0,i] - homotopy"
Definition;
The category
Tel
in Top N
into a filtered homotopy.
of telescopes is the full subcategory
consisting of telescopes.
This should cause no confusion. the telescope category
Tel;
Observe that the functor
also that
that cylinders may be formed within full subcategory of telescopes in
(3.7.4) Top N.
> Filt.
Then
Proof.
Proposition. Tel {f } n
Observe that
Let
Tel { X }
Tel.
Tel
x [0,i] - Tel {X n • [0,I])
We therefore let
Ho(Tel)
{f }:{X } n n
is invertlble in
Tel {X } n
9 {Y } n
be the
be a weak equlvalence in
Ho(Tel).
is a strong deformation retract in
Map (Tel {fn }) = Tel {Map (fn)}
= Tel {Xn x [0,i] u Yn/(X,l) - f(x)}.
{f } n
so
Ho(Fllt).
the mapping cylinder
We thus assume that
factors through
is also a coflbratlon in
Top N.
Tel
of
117
For each Hn:Yn x I
n
choose a retraction
>Yn
with
Hni0 = id
and
~ X,
rn:Y n
and a homotopy Caution:
Hnil ffi fn nr.
bond O r n + 1 # rnO bond,
in general
and
bond o Hn+ 1 # H n o bond.
However,
bond o rn+ I o fn+l = bond
(3.7.5)
ffi r
n
o f
n
bond
= rn o bond o f n+l'
(3.7.6)
bondoHn+ l o ( f n + l x id) ffi HnO bond o ( f n + l x i d ) :
We shall now use (3.7.5)
to define
Xn+ 1 x [0,I]
' Yn"
a filtered map
g;Tel
which will be shown to be a filtered-homotopy-lnverse map
fn+l
is a trivial
a strong deformation retraction.
and
cofibratlon,
retract
The composite
Yn+l
of
Yn+l x [0,I]
map,
to be denoted
to
{Y } n
Tel {f }. n
x 0 u Xn+ I x [0,i] for
n = 0.
Kn+l,
Tel {X }, n Because
u Yn+l x i
Let
Pn+l
the
is be the
118
Yn+l x [0,i] Pn--~->+~Yn+ 1 x O u X
(3.7.7)
n
x [0,i] U Yn+l x i
r n o b o n d u r n = b o n d o fn+l
u bond
o
rn+l~ X n
Yn+l -->
X I]
Xn+l
(see (3.7.5)) yields a homotopy from
r o bond
to
bond o rn+l.
Now define a
n
map
glTel {Y }
-,> Tel {X }
n
Xn+ I x [n, n +i]
to
X
x n
2t - n - l ) ;
according to the formula
g(y,t) ffi (Kn+l(y, 2 t - 2 n ) ,
Y
n
Yn+l x [n + 89 n + i]
n
(3.7.9) maps
maps
g(y,t) ffi (rn+l(y),
maps Yrrbl x [n, n + %]
g
g
according to the formula
(3.7.8)
and
as follows:
n
x n
to
X
x n
n);
according to the formula
n
(3.7.10)
g(y,t) = (rn(Y), n).
Yn+l
n
r n+l
Xn+ I
/
Y
n
r
tl
x n
I
to
119
Then
g
is a filtered map.
Claim 1:
The maps
this, first observe that
(3.7.5).
gf
and
K+llX+I • tO,l]
(3.7.11)
In
~ ~
by
n+~ ! t ! n+l,
~
g~(x,0 = ~I (x' L(~,n),
by (3.7.8) - (3.7.10).
Claim 2:
The maps
fg
and
Xn+ 1 x [ 0 , 1 1 x [ 0 , 1 ] '
tlon retract of
n < = t <= n+%,
Claim i follows easily.
idTe I {yn}
use (3.6.6) to imitate the construction of
interval.)
x+ 1
is t h e projection onto
Thus
2t
As ahove,
To c h e c k
are filtered-homotoplc.
idTel {X } n
g
are filtered-homotopic.
and obtain the required homotopy.
o Yn+l x 2 ( [ 0 , 1 ]
Y.+I • [ o , 1 ] x [ O , l ] ' .
x [0,1]')
([0,i]'
We may therefore define a homotopy
We shall
rn+1
is a strong deforma-
is just a second unit
from
bond rel Xn+ 1 x [0,I] x [0,i]' u Yn+l x a([0,1] x [0,i]')
Kn+ 1
to
schematically,
120
t bond
n
Gluing these maps together yields a filtered h o m o t o p y
r:Tel {Yn } x [0,i]'
from
fg
nares,
"to a map
h:Tel {Y } n
a n d moves t h o s e c o o r d i n a t e s
> Tel {Yn x [0,i]')
Tel {Yn }
which changes only vertical coordl-
a t m o s t 89 u n i t
analogous to those in Claim 1 and are omitted.
(3.7.12) Ho(TopN)
Corollary.
The functor
9 Tel (Yn }
Tel:Top N
as in
(3.7.11).
Details
Claim 2 follows.
> Ho(Tel)
are
0
factors through
to induce
Te1:Ho(Top N)
Proof. by a diagram
9 Ho(Tel).
By the model structure, each map in
Ho(TopN)
may be represented
121
{Xn}
(Yn)
{z' } I1
where
{Y' } n
is fibrant in
(levelwise) weak equivalence. Any two such m p s
Top N
(i.e., a tower of flbrations) and
(Note:
all objects in
-x[0,1].
We shall now extend
(3.7.13)
(Xn}
with
Tel {X } n
Top N
Tel
-x[0,1].
Further,
admits a homotopy inverse
~o Ho(Top, tow- Top).
Suppose that
(X
%
)
is a coflnal subtower of
Then there is a natural equlvalence
~ Tel {X } in Ho(Tel). nk
The r e q u i r e d map to
illustrated.
Xnk x k
Tel {X}
is a
are cofibrant).
The concluslon follows by Proposition (3.7.4).
Proposition.
Xno = x O.
Proof. Xnk x nk
N
are homotoplc with respect to the 6yllnder
a weak equivalence between fibrant objects in with respect to
Top
{in )
~ Tel (Xnk}
is d e f i n e d by mapplng
under the identity and extending to
Tel {Xn }
as
[]
122
X
~+1
x
• nk+ 1
X
•
This map is easily seen to be a filtered homotopy equivalence.
(3.7.14) a)
• (k+l)
k
[]
Proposltlon.
The functor
Tel:Top N
> Ho(Tel)
factors through
(Top, towyTop).
b)
This functor factors through
Ho(Top, two-Top),
Tel:Ho(Top, tow-Top)
Proof.
a map in
> Ho(Tel).
Part (a) follows from Proposition (3.7.13) because each map in
may be represented by a diagram
(Top, tow-Top)
where
to induce
{Xnk }
~s cofinal in
{Xn} , {Xno}= XO,
and
{fk }
is a level map (i.e.,
TopN).
Part (b) follows from Part (a) and the observation that
Y
is fibrant in n
123
(Top,
tow-Top)
if
{Y } n
is a tower of flbratlons by using the proof of
Corollary (3.7.12).
We are ready to show that
> Ho(Tel)
Tel:Ho(Top, tow-Top)
ylelds an
equivalence of categories.
(3.7.15) ~(X) = (X } n
Definition. in
The end of a filtered space
(Top, tow-Top).
Then
e:Tel
The definition of
(3,6,16)
Ho(Tel)
proposition,
c
Definitions.
be the map on
TopN
k > n.
extends to a functor
9 (Top, tow-Top).
induces a functor
Let
9 Ho(Top, tow-Top).
X ffi {Xn} e Top N.
given by lettlng
x [k-l, k]
for
,~
pnl~
Let
x [k-l, k]
bond o . . .
o bond
p:e o Tel (X) be the composite
~X n
Let
q ffi Tel (p):(Telo e)(Tel (X))
(3.7.18)
is the tower
implies the following.
~:Ho(Tel)
(3.7.17)
~
X
9 Tel (X).
Proposltlon,
a)
The maps
p
are natural weak equlvalences in
b)
The maps
q
are natural weak equlvalence in
Ho(Top,
Ho(Tel).
tow-Top).
9X
124
Proof.
Part (a) follows i~mediately from the definitions of
(note that each
Pn is an equivalence in
Ho(Top)).
Tel,
e,
and
p
Part (b) follows from
Part (a), the definition of the telescope category and Proposition (3.7.14b).
Propositions (3.7.14b),
(3.7.19)
Theorem.
(3.7.16), and (3.7.18) imply the following:
The categories
Ho(Top, tow-Top)
and
Ho(Tel)
are
naturally equivalent.
A geometric model for assume that all towers (Top, tow-Top) Tel,
ConTel
X = {Xn}
(regard
X
as
satisfy (*,X)),
We may
may be obtained as follows. X0 ~ * .
This embeds
tow-Top
in
and gives rise to a full subcategory of
(contractible telescopes) consisting of those telescopes
Tel (* = X0+--- ~ <
(3.7.20)
Ho(tow-Top)
.--).
Corollary.
The categories
Ho(tow-Top)
and
Ho(ConTel)
are
naturally equivalent.
(3.7.21)
Remarks.
categories of diagrams,
~x0 +-- xl<
...~.
In R. Vogt's [Vogt -I] approach to the homotopy theory of Tel {Xn}
is the homotopy collmlt of the diagram
In this setting our homotopy category
Ho(Tel)
represents
"coherent pro - hom~topy," that is, a version of pro - homotopy where maps and homotopies are required to satisfy various coherency conditions.
The development of
coherent pro -homotopy theory (Vogt only works with level maps) is an interesting problem whose solution should have applications to proper homotopy theory (see w and shape theory, especially in alternative proofs of the Chapman [Chap - i]
125
complement theorem (see w
Inj
w
spaces.
-
The above t h e o r y o f p r o - s p a c e s inj-spaces
Let
(direct
C
systems).
SS J~
(3.8.1)
C
indexed by
to
theory without proofs.
> Yj})
f
f. 3
in C J~
respect to all maps
(3.8..2)
C J~
is the category of
The closed model structure on the level
TheOrem
[ B - K , p. 314] (dual to (3.2.1)).
in C J~
the maps
A map
J.
Then
Let
is due to Bousfield and Kan [B- K, p. 314].
Definitions
(= {fj:Xj in J,
sketch this
be a cofinite strongly directed set.
direct systems over
j
We s h a l l
dualization
be a closed model category which satisfies Condition N of w
J ( = {j})
category
admits a straight-forward
A map
f:X--~Y
is a fibration (resp., weak equivalence) if for all
are fibrations (resp., weak equivalences) in
C.
is a cofibration if it has the left-lifting property with p
which are both fibrations and weak equivalences.
(dual to (3.2.2)).
C J~
together with the above structure
is a closed model category.
(3.8.3)
Proposition
cofibration if for each
(dual to (3.2.3)). j
in J
A map
the induced map
f:X--->Y qj
in C J~
in the diagram
is a
126
>X. ] \
1 c o l i ~ < J fkl
c~
< j Yk
9-
>Pj~,\\~ \
J (Pj
is the pushout) is a cofibrat!on.
This result, in the ease CW
J = N,
is the usual definition of cofibration of
spectra (see [Vogt -2]).
(3.8.4)
Theorem
(dual to (3.2.4)).
The constant diagram functor
preserves cofibrations, flhrations, and weak equivalences. functor
colim:C J~
>C
C---->CJ~
The direct limit
preserves cofibrations and trivial cofibrations (maps
which are both cofibrations and weak equivalences).
We shall now discuss the homotopy theory of
(3.8.5)
stron$ cofibration if in some
(dual to (3.3.1)).
Definitions
cJ~
where
f J
is the image in
inJ -C.
A map Inj -C
f in inj -C
is called a
of a (level) cofibration
is a cofinite strongly directed set.
{fj}
Strong fibrations,
stron$ trivial cofibrations, and stron~ trivial flbratlons are defined similarly.
127
A map in
inj -C
is a cofibratlon
of a strong cofibration.
Fibrations,
in
inj -C
are defined similarly.
if
f = pi
where
(3.8.6) structure,
Theorem
p
if it is the retract in
trivial cofibrations, A map
f
in inj - C
is a trivial fibration and
(dual to (3.3.3)).
i
inJ -C,
Maps(inj -C)
and trivial fibrations is a weak equivalence
is a trivial coflbration.
together with the above
is a closed model category.
(3.8.7) C--->inj - C
Theorem
(dual to (3.3.4)).
preserves cofibratious,
direct limit functor
eolim:
The constant diagram functor
fibratlons,
inj -C -->C
and weak equivalences.
preserves cofibrations
The
and trivial
cofibrations.
The results of w167 note for later reference
- 3.7 may also be dualized to that the full subcategory
of spaces indexed by the natural numbers)
contained
inj - C.
of direct towers in
Ho(inj -C)
We shall simply (direct systems admits a
geometric model along the lines Of w
(3.8.8) direct tower
Definition
(Milnor
X = {X } n
is the space
Dir tel (X) = X 0 x [0,i] u bond
0
[Mil -3], dual to (3.7.2)).
X 1 x [1,2] u bond "'"
I
2
The telescope of a
128
Dir tel (X) tion).
admits a natural increasln~ filtration
D
We shall now consider those direct towers tractlble direct telescopes inj -C
(we call this a cofiltra-
to a direct tower
(3.8.9) equivalence
contractible
Theorem
(agaln,
direct
X 0 = *).
X ffi {X } n
with
(dual to (3.7.20)).
of homotopy categories
X ffi {X n}
from
with
X 0 ffi *,
and con-
Any direct tower is equivalent in
X 0 ffi *.
The direct telescope functor induces an Ho(inJ - C)
t e l e s c o p e s and c o f i l t e r e d
maps.
tO the homotopy category of
4.
THE HOMOTOPY INVERSE LIMIT AND
ITS APPLICATIONS TO HOMOLOGICAL ALGEBRA
w
Introduction. In this chapter we shall define a homotopy inverse limit functor
holim:Ho(pro - C)
~ Ho(C)
adJolnt to the inclusion functor
for suitable closed model categories study of the derived functors
lim s
C
(w
Ho(C)
~ Ho(pro- C)
and obtain applications to the
of the inverse limit.
Bousfield and Kan
[B -K; Chapter XI] gave a less intuitive construction of a (somewhat different) homotopy inverse limit on
pro -SS J
(see w167
4.9) j and suggested the study of
homotopy inverse limits on other pro - categories.
In w
w
is an appendix to w
we survey some of the results of other authors, notably Z. Z. Yeh,
J. -E. Roos, J.- L. Verdier, C. Y. Jensen, and B. Osofsky, on for our own work. sections: limS;
w
w
Our results are stated in w Algebraic description of
Vanishing theorems for
In this chapter, Condition
N
(w
shall discuss abelian groups).
C
limS;
as background
and proved in the following w
Topological description of
lim s.
shall denote a closed model category which satisfies
and admits arbitrary (inverse) limits.
C = SS,
llm s
SSG
(simplicial groups) and
SSAG
In particular, we (simplicial
130
w
The homotopy inverse limit. In this section we shall define a homotopy inverse limit functor
holim:
adJoint to the inclusion {Yj}
in
Ho(C) "
Ho(pro -C)
~Ho(C)
9 Ho(pro -C).
That is, for
X
in C
pro-C,
(4.2.1)
Ho(pro-C)(X, {Yj}) i Ho(C) (X, holim {Y.}).j
We begin by considering formula (4.2.1) in the case that and hence in
(4.2.2)
pro -C,
and
Proposition.
(Yj}
is fibrant in
X
is cofibrant in
pro -C.
The inverse limit functor
lim:(pro -C)ef
'C
induces a functor on the homotopy categories
So(C).
lim: Ho((pro -C)cf)
Proof.
To define
lim
on maps in
Ho((pro -C)cf),
recall that for
{Xj}, {Yk } E (pro-C)cf,
(4.2.3)
Ho((pro -C)cf)({Xj},{Yk}) = [{Xj},{Yk}],
the set of homotopy classes of maps from cylinder pro -C, Hi I = g.
and
{Xj} ~ and let
[0,i]. H:{X.} ~ 3
Let [0,i]
f,g:{Xj } ~Y
{Xj}
to
~ {Yk }
{Yk }
with respect to a
be homotopic maps in
be a homotopy with
Applying the inverse limit to the diagram
Hi 0 = f
and
C,
131
{X.} 3
[X.} |
(4.2.4a)
3
[0,1]
H > {yk}
{x.} 3
yields the diagram
lim {Xj }
k~'~O
lim {Xj
.} ~S
i
Now observe that the maps trivial fibration
lira ({Xj} ~) [0,I])
lim H) lim {Yk }.
I
i0 and iI
p:{Xj} ~
[0,i]
in diagram (4.2.4a) are sections to the ~ {Xj}.
Hence,
lim i0
and
lim iI
are
sections to the induced (lim preserves trivial fibrations, see Theorem (3.3.4)) trivial fibration the maps
lim i0
lim p: lim ({Xj} | and
lim iI
[0,I])
> lim {Xj}.
are weak equivalences.
This implies that
By diagram (4.2.5), the
maps lim f = lim H o l i m
i0
and lim g = lim Ho lim i1 become equivalent in
(4.2.5)
Ho(C).
Proposition.
D
The inverse limit functor
a functor on the homotopy categories,
lim:(pro -C)
>C
induces
132
llm:
Proof.
Let
Ho((pro -C)f) ----~Ho(C).
{Xj) ~ pro -C.
Factor the natural map
~ --~{Xj)
a cofibratlon followed by a trivial flbratlon.
~o(pro-c)({xj},-) = flbratlons,
Ho(pro -C)({~k},-);
also, because
Ho(C)(lim { 5 },-) ~ Ho(C)(lim {~},-).
lim
as
Then
preserves trivial
The conclusion now follows
from Proposition (4.2.27.
(4.2.6)
Proposition.
For
X
cofibrant in
C
and
{Yj)
flbrant in
pro - C,
(4.2.7)
Proof.
Ho(pro -C) (X, {Yj}) e Ho(C) (X, llm (Yj}).
Formula (4.2.7) follows easily from adjolntness of
llm on pro - C,
p r o - C (X, {Yj}) = C (X, lim (Yj}),
and Proposition (4.2.2).
[]
We are now ready to define the homotopy inverse limit. model categories (see w Ho((pro- C)f)
the homotopy theory of flbrant objects in
is equivalent to the homotopy theory of
Following Quillen, for each object a trivial eoflbratlon
ix
(4.2.8)
(If
X
is flbrant, choose
By Qu/llen's theory of
X
in pro - C,
pro- C p
factor the map
pro- C~
Ho(pro - C). X---~*
followed by a fibratlon
X iX
Ex~X = X).
Ex~X
9 *
This construction induces a functor
as
133
(4.2.9)
Ex~:Ho(pro- C)
9 Ho((pro -C)f);
is defined on morphisms by applying Axiom M6 to obtain fillers in the diagrams
f
X
> Y
Ex X ............ ~ Ex=Y
The homotopy class of
Ex~f
depends only on the homotopy class of
f
because
[Ex~f] = [iy] [f] [ix ]-1.
We therefore define the homotopy inverse limit, holim, to be the composite functor
(4.2.10)
holim ~ llmo Ex ~ : Ho(pro -C)
(4.2.11) (4.2.8).
Remarks.
Hence,
Propositions
(4.2.12) the inclusion
If
> Ho((pro -C)f)
X ~ pro- C
holim X = lim X
on
> Ho(C).
is fibrant, we may take
Ex~X = X
in
(pro -C)f.
(4.2.5) and (4.2.6) im~edlately yield the following.
Theorem.
The functor
Ho(C)
~ Ho(pro - C).
holim: Ho(pro -d)
~ Ho(C)
is adjoint to
134
(4.2.13)
Remarks.
Bousfield and Kan [B -K, Chapter XI] defined a different
"homotopy inverse limit" functor that for a tower of fibrations fibrations
X,
we may take
ours on suc h systems.
holimB_ K : Ho(SS J) X,
* Ho(SS).
holimB_KX ~ llm X.
Ex=X = X;
They observed
But, for a tower of
hence their definition is equivalent with
In general our definitions differ except on fibrant objects.
For example, Bousfield and Kan only obtain the following analogue of Theorem (4.2.11):
R holimB_ K
R holimB_ K
is adjoint to the inclusion
Ho(SS) ----*Ho(SSJ) j
is Quillen's total right deriVed f u n c t o r [ Q
In fact, for our
holim,
holim = R holim = R lim,
-i, w
where
lim
of
where
hol~_
K .
is the ordinary
inverse limit.
w
Ex ~
o_~n pro- S~S.
In this section we shall describe an explicit to
Ho(pro -SS)f,.
X--*Ex~X
in
(4.3.1)
together with natural
pro -SS.
Ex |
o~n objects,
MX=
{Xj~
maps
Zj
X. --** J
He(pro -S$)
trivial cofibrations
Let
X ~ pro- SS. X
First apply the Mardew
(Zj} = Ex~{X.) J
con-
functorially by a naturally isomor-
where
J = {j}
J = {J}.
is a cofinite strongly
We shall proceed inductively.
First, suppose that cial sets
Ho(pro ,SS))
indexed by a cofinite strongly directed set
We thus need only define directed set.
functor from
Compare (4.2.8) - (4.2.9).
struction (Theorem (2.1.6)) which replaces phic object
(in
Ex |
J
is an initial object of
and a trivial cofibration as in [Q-I~
w
X.J --*Zj
J.
Define a flbrant simpll-
by functorlally factoring the
135
Next, assume inductively that for a given simplieial sets brations
a)
~,
~--->~
for
bonding
maps
j
in J, for
Zk ----+ZA
and all
s < k,
k < J,
and trivial cofi-
have been defined so that:
s < k
the diagrams
> Yk
XA --
> Ys
commute;
b)
for
m < s < k
the diagrams
Zk - -
commute;
c)
; Zg.
and
The maps
Zk
> lims < k{ZA}
induced by (b) are fibrations.
Apply the Quillen factorization to the composite mapping
xj to obtain
a diagram
>lirak <S {Xk}
~ lirak < j{zk}
that
136
Xj
, Zj
> lisk< j{Zk}
The map
consisting of a trivial cofibration followed by a fibration. Zj
> iimk< j{Zk}
induces bonding maps
Zj --+Z k
k < j.
for
It is easy to
Condition (c) above is
see that these maps satisfy conditions (a) and (b) above. satisfied by construction.
Continuing inductively yields a fibrant pro- (slmplicial set) By construction,
Ex~
extends to a functor on
In order to define
Ex =
on morphisms in
S$J.
pro- SS,
{Zj} = Ex={Xj}.
D
we need the following
cofinality lemma.
(4.3.3)
Lemma.
directed sets. EX={~:}__~
Let
T:J --->K
Then for any
> Ex~{X~(j)}z
in
{~},
be a cofinal functor on cofinlte strongly T
induces an isomorphism
Ho(pro -SS),
SS K
in other words, the diagram
T
~ SSJ
S K
SSJ
\
/ tto (pro - SS)
commutes up to natural equivalence of functors.
Proof. assume that
Let
{Yj}
J and K
denote
Ex={XT(j)}
and
have initial elements
{Zk} J0
denote
and k 0
with
Ex={Xk}. T(J0) = k 0,
We may and'
'i37
also that
~ 0 ffi*"
j' in J,
and assume that the natural maps
j < j'
This yields a natural map
ZT(J0 )
ZT(j) ---->Yj
+ Yk0.
Now fix
have been defined for
such that the diagrams
(4.3.4)
ZT(j)
, Z.3
XT(j)
> ZT(j) ",.
and
ZT(j,,)
commute.
:- zj,,
Y. 3
Consider the induced commutative diagram
(4.3.5)
> l ~ k < T(J'){~}
~ limk < T(J'){Zk}
limT(j)
xr(j,)
Since
ZT(j, )
II
> limj <j,{~(j)}
limT(j) < T(j'){ZT(j)}
is defined by applying the Quillen factorlzatlon to the composite
along the top row Of diagram (4.3;5), namely the map and
Yj,
1
>limj< j,{Yj)
~(j,) ---+ limk< T(j,){Zk},
is defined by applying the Quillen factorization to the composite along
the bottom row f (4.3.5),
~(j,)
) limj < j,{Yj},
there is an induced map
138
(4.3.6)
ZT( j ,) ---->Yj, .
By diagram (4.3.5); the map (4.3.6) satisfies the conditions of diagrams (4.3.4). Continue inductively to define maps
ZT(j)
~ Yj
(4.3.4) then yield a commutative diagram in
(4.3.7)
{~(j)}
for all
j
in J.
Diagrams
SS J
~ {ZT(j)}
=
T*Ex~{~}
3
Since for each
j,
~(j)
= ZT(j)
and
~(j)
= Yj,
level weak equivalence, hence an isomorphism in
the map
z
in (4.3.7) is a
Ho(pro -SS).
Since the constructions which led to diagrams (4.3,7) were natural, we have defined the required natural isomorphism
~~
~ -" ~ ~ 1 7 6
=
We have thus proven that the construction
' ~~176
X
*.
) MX
D
~ Ex=MX,
denotes the Marde~id construction, extends to a functor from Ho((pro - SS)f)
and that there are trivial coflbratlons
which are natural in Ho(pro -SS),
(4.3.8)
Ho(pro -SS).
By construction,
pro- SS
X _z MX ExaM
where to
>Ex MX, factors through
as required.
Remarks.
The only special property of
existence of functorial factorizatlons in Axiom M2 (w
SS
which we used is the
139
w
The derived functors of the inverse limit:
background.
In this section we shall briefly summarize vanishing theorems and cofinality theorems for right-derived
functors
lim s
lim:
(where
AG
of the inverse limit
(AG) J
> AG
is the category of abelian groups).
Because
no___~t,in general, right exact, vanishing theorems for of
lim.
Cofinality theorems are used to extend
lim
lim s lim s
is left-exact, but measure the exactness
to
pro -AG.
There
is a close relationship between these results and the homological dimension of modules which we shall not discuss here; see B. L. Osofsky
[Osof] for a good survey.
We refer the reader tO [Mit -2], for example, for the basic theory of derived functors due to H. Cartan and S. Eilenberg
[C -E].
Z. Z. Yeh [Yeh] gave the first vanishing theorem in 1959.
His results were
extended first by J. -E. Roos [Roos] in 1961 and later by C. U. Jensen [Jen] in 1972.
(4.4.1) directed set
Definition I
(4.4.2) limS{G i} = 0
set
J
> limi~ j{G i}
Proposition for
[Roos]).
is called flasque
necessarily directed) limi E l{Gi }
([Yeh],
s > 1.
An inverse system
(or star-epimorphic)
contained in
I,
{G i}
indexed by a
if for each ordered
the natural map
is surjective.
([Yeh],
[Roos]).
If
{G i}
is flasque,
then
(not
140
Roos obtains this result by showing that complex obtained from
{Gi}.
(4.4.3) set
I
I
CW
complexes.
([Jen]).
An inverse system
lim i ~ l{Gi}
Proposition
l ims{C i} = 0
for
~ lim i ~ j{G i}
([Jen]).
lim
If
indexed by a directed J
contained in
I,
the
is surjective.
{G i}
is weakly flasque, then
s ~ i.
Jensen first proves that showing that
liraI
and
In fact, Roos only
{Gi}
is called weakly flasque if for each directed set
(4.4.4)
lim
be ordered, not necessarily directed.
Definition
natural map
is the homology of a
Compare J. Milnor's [Mil -3] use of
in axiomatizing the cohomology of infinite requires that
lim s {G i}
lim s
applied to a weakly flasque system is
is rlght-exact on such systems.
Vanishing of
lim s
0
by
then fol-
lows by a suitable iteration.
(4.4.5)
a)
Remarks.
Clearly flasque implies weakly flasque.
Jensen observed that
the converse is false.
b)
Roos and Jensen actually worked in categories of the form where
A
A I,
is an abelian category which satisfies suitable
exactness axioms of A. Grothendleck [Gro -2].
One may extend the domain of follows.
First, a cofinal functor
lim
and
lim s
T:J --->K
from
(AG)J
to
pro- AG
induces an isomorphism
~S
141
T*:{G k}
~ {GT(j)}
in pro ~ AG,
hence we need the following result of Roos
(1961), Jensen (1972) and B. Mitchell [Mit -2] (1973).
(4.4.6)
Theorem [Mit-2].
A coflnal functor
T:J--~K
between filtering
categories induces commutative diagrams
K
l~KS
(AG
AG
T*I (Aa J Secondly, although Artin and Mazur [A -M, w representation of a map {Gj(s
---~(s
{Nilj E J}
in some
level representatives.
(AG)L,
~ { ~ I k c K}
J (~)
where
commutative diagrams
in pro -AG
a given map in
}
pro -AG
by a level map may have many
~ {~(~) }
{Gj(m) } "'"
pro- AG,
above) gave a natural
Therefore we need to show that commutative diagrams
{G.
in
(see w
L = {s
and
" {Hk(m) }
M = {m}
are filtering categories, induce
142
I~Ls(cj (~) }
.
~
~ li-Ls{Hk(~) }
S
IL'~N {C3(m) }
- 1~s{~(~)
}-
J. -L. Verdier [Ver] announced this result in 1965, and hence extended the inverse limit functor and its derived functors to
pro - AG.
limj (AG) J
In particular, the diagrams
s
o.
> AG
pro - AG
commute, where
limproS
is the s th right derived functor of
limpr ~ .
We
shall include independent proofs of these results, see w
(4.4.7)
lim and HOM functors.
the category of R - m o d u l e s . R -Mod,
that is,
underlying set. the functors R ~
J
Let
HOM (X,Y)
Let HOM
R
be a commutative ring, let
R-Mod
be
denote the internal mapping functor on
is the natural R -module with
R - Mod (X,Y)
as
In 1973 B. L. Osofsky [Osof] gave the following representation of limS:(R-Mod) J
~ R - Mod.
such that the categories
(R- Mod) J
She defined a "tensor product" ring and
(R~
This isomorphism induces natural equivalences of functors
J) - Mod
are isomorphic.
143
(R -Hod) J
lira:
~ R -Hod
(~ HOM(R _Mod)J(R, lim(-)):
(R -Mod) J
i HOM,R~ (9 j _Mod,(Rj (9 I,-):
> R - Mod)
R (9 G -Mod
9 R- Mod,
hence also
liras:
(R -Rod) J
9 R - Mod
s EXt(R (9 J ) - Mod (R (9 J , - ) :
(R (9 J-MJod)
7R-Mod.
Later in this chapter we shall use pro- categories to give more natural results See w
w
Results on derived functors of the inverse limit.
We shall prove the following three theor~s in w
Theorem A. HOHpro,
Extj
Let and
J
be a cofinite strongly directed set, and let
EXtpr ~
be the appropriate
a)
limj - HOMj(Z,-) : AG J - - A G .
b)
limpr ~ = HOMpro (Z ,-) : pro -AG --~ AG.
c)
limj s ~ ExtjS(z,-):
d)
llm
pro
Theorem B.
s =
Ext
Let
pro
J
HOH
and
Ext
HOHj,
functors.
Then
AGJ--->AG.
S(z,-):
pro -AG--~AG.
be a cofinite strongly directed set.
Then the diagrams
144
AGJ
~
limjs~
>
pro
~ i ~
-
AG
pro
s
AG commute up to natural equivalence.
Theorem C.
Let
{G.} be a stable pro- group.
-
Then
3
{Gj}, " s = 0, llmjS{Gj } = limproS{Gj} -= limpr Os l i m {Gj}) ~ f[ lim 0, s r 0 .
In w
we shall use the relationship between the topological and algebraic
structures on
Theorem D. {~_} ~ p r o - AG.
pro -SSAG
Let
to prove the following.
{Gj} be an inverse system of free abellan groups and
Then
Ext s({Gj }, {Hk}) = Ho(pro - SSAG) ({Gj }, WSHk})
where
is the W-constructlon of Ellenberg and MacLane (see (4.7.9)).
Theorema A and D imply the following.
Theorem E.
limS{Gj} = Ho(pro-SSAG)(Z, {~Gj}) = w0{~Gj}
An analogous formula holds on
Theorem F.
lims{cj }
of slmpllclal spectra and
=
pro-G
for
s = 0
Ho(pro -Sp)(sO,{KGj}), KGj
and
where
on
pro-AG.
1.
Sp denotes the category
the simpllclal Ellenberg-MacLane spectrm~ with
145
Gj,
n = O,
0,
n # 0 .
~n(KGj) =
Bousfield and Kan [B - K, w
In w
give an analogous formula.
we describe our [ E - H -5] strong-Mitta~-Leffler
groups, a natural extension of the Mittag-Leffler [E - H -5] we proved that a pro -group if
G
is pro -isomorphic
directed set.
Let
a) liml{Gj }
= O;
c)
w
If
{Gj}
condition for towers.
is strongly-Mittag-Leffler
{Gj}
is abelian,
lim {G.} = O, 3
then
then
limS{Gj} = 0
{G.} ~ 0 3
in
for
pro - group.
limpr ~
pro - A G
extends
derived functors (4.4.7).
limj, Ext s
Then:
s > 0;
pro - G.
o_ff lim s.
In this section we shall interpret the inverse limit functors lim : pro
if and only
proof of the following in w
be a strongly-Mittag-Leffler
Algebraic description
and
In
to a flasque pro -group indexed by a cofinite strongly
We give a "topological"
Theorem G.
b) If
G
condition for pro -
> AG
as suitable
HOM
functors.
> AG
This will-show that
and also identify the derived functors of HOM.
llmj:AG J
lim s
as the
Compare the results of B. Osofsky described
in
Our results can be extended easily to categories of modules over a tom-
mutative ring with identity.
We shall need the following structure.
146
(4.6.1)
Theorem.
a)
AG J
is an abelian category.
b)
pro- AG
is an abelian category.
Part (a) is well-known.
For part (b), see [A -M, w
Some of the abelian
structure is described below.
(4.6.2)
Definition.
is exact if for each
j
A sequence in J,
0
~ {Aj} ~
the sequence
0
{Bj} >A.3
>{Cj} ~Bj
~0 ~Cj
in AG J ~ 0
is
exact.
(4,6.3) pro - AG
Proposition.
A sequence
0
~ {Aj}
> {Bk}
~ (C A)
0
in
is exact if and only if there exists a commutative diagram
0
~ {Aj } --~ {Sk}
--- {C~} - ~ 0
0
> {A m} ---~ {Bk(m) } --> {C' m} --~ 0
(4.6.4)
in which the bottom row is a short exact sequence i n the appropriate level category
A~,
M = {=}.
Proof.
The "if" part is clear.
For the "only if" part, first relndex the
given short exact sequence to obtain a diagram
147
0 --> {Aj }
---> {Bk}
--'>{C A}
--~0
(4.6.5)
0 --->{Aj (m) } --> {Bk(m) } --> {CA(m)} ---->0
in which the rows are exact, and the bottom row consists of level maps. M = {m},
and let
{A'm}--->{Bk(m)}
A'm
be the kernel of the map
is a kernel of the map
Bk(m) ~
CA(m).
{Bk(m)}--->{CA(m)}
Let Then the map
in pro-AG,
hence
we may replace the bottom row of diagram (4.6.5) by the isomorphic short exact sequence
(4.6.6)
0---> {A'm} ---~{Bk(m)}--~{Cg(m)}--'~0.
Similarly, let
C'm
be the cokernel of the map
short exact sequence in
A'm --~ Bk(m)"
There results a
AGM,
0 --> {A'm} -'->{Bk(m) } --->{C'm} ---~0,
which is isomorphic to (4.6.6), via reindexin8 in the middle term, as required.
(4.6.7)
Remarks.
Proposition (4.6.3) can easily be extended to finite dia-
grams o f s h o r t e x a c t sequences w i t h o u t
(4.6.8)
Definitions.
define direct sums in J
and K
0
Direct sums in
pro - AG
respectively; let
loops;
AG J
as follows. {Xj} ~
see w
are defined degreewlse. Given
{Yk } ffi {Xj
~
{Xj} Yk },
and
{Yk }
We may indexed by
indexed over the
148
product category if
J x K.
and
j ~ j'
J
If
and
(J,k) ~ (j',k')
HOM
on
AG
also extends to
pro-AG.
(4.6.9)
Definitions.
Given
HOMj({Xj},{Yj}) = AGJ({xj},{yj}), Similarly, given
{Xj}
HOMpro({Xj},{Yk})
Then limit
are directed sets,
k ~k'.
The internal mapping functor AG J
and K
the incluslon HOM (Z,X) m X
(4.6.10)
for
X
in AG,
b)
l i m p r o = HOMpro(Z,-):
(4.6.11)
Corollary.
(4.6.12)
Corollary.
limpro:
A G - - ~ p r o - AG),
limj s - ExtjS(z,-):
b)
lira
s = Ext
pro
AGJ --->Ag. pro - A G - - > A G .
B
AGJ--*AG.
AGJ--->AG.
S(z,-):
pro-AG--~AG.
{Yk }-
Because the i n v e r s e
pro -AG--~AG)
we have the following.
llmj = llmpr ~ . w:
a)
{Yj}.
let
Theorem.
l i m j ffi HOMj(Z,-):
define
with group operations induced from
(respectively,
a)
pro
in AG J,
may be extended to f u n c t o r s .
(respectively,
AG--~AG J
{Yj}
{Yk.} in pro -AG,
HO%r ~
' ~ AG
and
With group operations induced from
= pro -AG ({Xj),{Yk)),
HOMj and
limj:AG J
and
{Xj}
and
is adjoint to
149
Proof. isomorphic functors
In each case, functors.
llm s
and
[Mit -i,
p. 193D.
The above theorem and corollaries natural transformation
of connected
sequence) AG J
in
AG J
of
form Theorem A.
T:{ExtjS(z,-)}
~ .
into an extension in
natural transformation
in
Let
J
9
T:{Extj s}
~ {EXtproS},
induces an isomorphism
---->{EXtproS(Z,-)}.
they map an extension
p r o - AG,
p r o - AG.
be a coflns
AG J
follows.
We shall now define a
~
(long exact
and send a map of extensions
Therefore,
in
T induces the required
of connected sequences of functors
Theorem.
of derived
Because the natural quotient functors
preserve abellan structures,
into a map of extensions
(4.6.14) diagrams
The conclusions
is a coflnite directed set, then
COnstruction
~ pro -AG
characterization
of connected sequences of functors
sequences of functors
(4.6.13). ~:AG J
J
are the s th derived functors of
Now apply D. Buchsbaam's
[Buch] (see also
and show that if
Ext s
:{Extjs}
strongly directed set.
,
{tproS}
Then the
> pro - A G
ExtjS(z,-)~ ~XtproS(Z,-) AG commute up to natural equivalence.
Proof.
For
s = 0,
now follows by Buchsbaum's
this follows from (4.6.10) - (4.6.13). characterization
of derived functors
The conclusion
[Buch], or by the
150
following alternative direct proof for
We shall first show that
s > 0.
T:ExtjI(z," {Gj})
epimorphlsm for each cofinite strongly directed set Consider an extension in
(4.6.15)
replace the monomorphiem > {H'~},
we may assmne that
and L
for each
{J(~(j))}
{Gj} --->{Hk}
in
AG J .
j
Now use the maps
(4.6.18)
Finally, the map
as well as
J
L ffi {s
is a First,
{C~}
to obtain
{Cs
is a coflnite strongly directed set.
are eonflnite strongly directed sets, we may choose elements in J
is cofinal in
(4.6.17)
K
by an inverse system of monomorphisms
O--->{Gs163
Again, we may asstmse that
s
{Gj}
We shall relndex (4.6.15) several times.
and take the levelwise cokernel
(4.6.16)
J
and
o --~ {G~ } - ~ {R k} --+ Z - ~ O.
coflnite strongly directed set.
Because
J
pro- AG
By the Marde~id construction, w
(Gj(~)}
is an
> EXtprol(Z, {Gj})
so that J.
j ~J'
implies
~(j) ~ s
and
We now replace (4.6.16) by
0 --+ {G.j(~(j))--~ {H'~(j) }-~
Gj(~(j)) ---~Gj
{C~(j)}-->0.
to push out (4.6.17) and obtain
O--->{Gj}--+{H"j}--+{C'j}--->O.
Z a {Z.3 = Z}
~ {C'j}
is a pro-isomorphlsm, so pulling back
(4.6.18) by this map yields the required extension
151
(4.6.19)
0 --+ {G.} --+ {H"'j) --->{Z.} --+ 0 3 3
isomorphic to (4.6.15). epimorphiem. set
J
and
T:ExCjI(z, {Gj}) --+EXtprol(z, {Gj})
is an
Similar techniques imply that for each cofinite strongly directed (Gj} in AG J,
isemorphism for all system.
Hence,
s.
T:ExtjS(z, {Gj}) ---->EXtproS(Z, (Gj }) The crucial
point
is
that
Z
is
is
a constant
an
inverse
D
Because~ we have already identified
lim s
as
Ext s,
TheOrem (4.6.14)
implies the following.
(4.6.20)
Theorem B.
Let
J be a cofinite strongly directed set.
Then the
diagrams AGJ
~
limj~s~
+ pro - AG /improS
AG
commute up to natural equivalence.
Hence we shall write
lims
(4.6.21) Definition. it is isomorphic in
If pro- G
the
by functoriality of
ing vanishing theorem.
limjs = llmpr ~ s
An inverse system of groups
pro- G
{G.}3 is stable,
for
(If
{Gj}
is called stable if
to a group.
natural map lim.
lim {Gj} --+~G.}3
is an isomorphism in
Theorem B then immediately implies the follow-
{G.} i s not abelian, everything works for 3
152
s = 0
or
i.
non-abelian groups.
(4.6.22)
[B -K]
See, for example, In this case,
Theorem C.
Let
lim I
{Gj}
for
Topological description of
of an inverse system of
is only a pointed set.)
be a stable p r o - group.
limaS{Gj} = l~mproS{Gj} = 1 r o S ( l i m i m
w
llm I
{G.}) ~ 3
{
li=
Then
{Gj},
e = 0
O,
s#O.
llm s.
In the last section we showed that
= Extl(z, {Gj })
lim I Gj
where
J ffi {j)
is a cofinite strongly directed set.
Because
ExtI(z, {Gj})
is the set of short exact sequences
0 ---* {Gj } ~
and a short exact sequence in
{Sj } ~
pro - AG
{Z} ---~ 0,
is a fibration sequence in
pro -SSAG
(see (4,7.1) - (4.7.13), below), one is led to ask whether
Extl(z, {Gj}) = Ho(pro-SSAG)(Z,B {Cj}) for a suitable classifying space in this section.
abelian group
We shall carry out the above program
The first step is to relate the abeliau structure of
and the closed model structure of
(4.7.1)
B{Gj}.
Definition. SG
with
pro - SSAG.
Associate to an abellan group (SG) n = G
p r o - AG
for all
n ~ O,
G
the dlscrete simplicial
and all face and degeneracy
153
maps the identity. TH = H0,
Associate
to a simplicial abelian group
the group of 0 - simplices of
(4.7.2)
Proposition.
Then
S
and T
coadjoint to
T,
S
extend to functors
9 SSAG,
T: SSAG
S
the abelian group
H.
S: AG
with
H
9 AG,
a full embedding,
and
TS = lAG.
The proof is easy and omitted.
Now prolong
S
and T
to functors
(4.7.3)
S: pro -AG----~pro - S S A G T : pro - SSAG
(4.7.4) TS = i
Proposition.
is c o a d j o i n t to
S
Immediate from Proposition
We shall frequently identify
A
T,
S
iS a full embedding and
p r o - AG.
Proof.
S.
9 pro - A G
Artin and Mazur
[A -M,
is an abellan category.
w
(4.7.2).
pro - AG
with its image in
showed that
See w
pro - A
for the case
pro - SSAG
under
is an abelian category if
A = AG.
We are in the fol-
lowing situation.
(4.7.5)
kernels,
SSAG
is an abelian category.
c o k e r n e l s , and d i r e c t
The required structures,
sums a r e d e f i n e d d e g r e e w i s e .
namely
Addition in
0,
154
SSAG (G,H)
is defined degreewlse.
The functors
S and T
(4.7.2) preserve
abelian structures.
(4.7.6)
The normalizatlon
is the chain complex
NG
of a simplicial abellan group
NG = {NnG,dn}
I NnG =
G - {Gn,dnl,snl}
with
G 0, (dni:Gn 0 ker i>0
n = 0 n = 0
~ Gn_l),
dn = dnoINnG.
Then
N
extends to a functor on
SSAG.
Moore showed that
w,G = H,{NnG,dn),
the homology of the chain complex
(4.7.7)
Call a simpllclal abelian group
S:AG--->SSAG, 0.
If
dnl = 1G0
G
NG.
is a discrete simplicial abelian group, for all
n and i;
G
G
n
= GO
is in the image of G
have dimension
for all
u,
and
hence
I G0,
G
discrete if
that is, if the only non-degenerate simpllces of
NnG
if
G
n ffi0
=
O,
n#O
is discrete.
Proposition (4.7.6) ylelds the following relatlonship between the abelian and closed mode I structures of
SSAG.
~S5
(4.7.7)
PropOsition
a flbratlon in
(Quillen [ Q - I , Proposition II.3.1]).
SSAG
(hence in
SS)
if and only if
N f n
A map
f:G --~H
is surjectlve for
> 0.
(4.7.8)
Remarks.
We shall need two special cases:
a)
An 7 map of discrete simpllclal abelian groups is a fibratlon;
b)
Any (levelwlse) surJection of simpllclal abellan groups is a flbratlon.
The fibre of a fibration
(4.7.9) SSG
f
in SSAG
is Just the (levelwlse) kernel of
f.
S. Eilenberg and S. MacLane defined a classlfying space construction
(see [May - 1, p. 21] for a description).
To a slmpllclal abellan group
their construction associates a flbratlon sequence in
SSAG
G--->wG--->WG;
is a contractible (in Lct
WG
SSAG) simpllclal abellan group, and
is always a slmpllcial group, even if
G
WG ~ SSAG.
is not abelian.
In
Their con-
:ruction Immediately yields the following.
(4.7.10) l
SSAG
W
takes a short exact sequence
into a short exact sequence
(4.7.11) ~ly if
Proposition.
Wf
Proposition. is a flbration.
A map
0 -~K
f:G--~H
0 --~ K --~ G --~ H --~ 0
--~WG --~WH --> 0
in SSAG
in SSAG.
is a surJectlon if and
156
Proof.
The "if" part follows from Proposition (4.6.10) and Remarks (4.7.8).
For the "only if" part, the map
(WG) 0
NWf:NWG--~NWH
W.
(~)0
each consist of trivial group
0,
is a degreewise surjection by Proposition (4.7.7).
[Q -I, Lemma II.3.5], the map the construction of
and
Wf
is a surJectlon.
Hence,
f
so By
is a surJectlon by
0
We shall now extend the above discussion to
pro - SSAG.
Call a pro-
(slmplicial abelian group) discrete if it is pro - isomorphic to one in the image of S:pro -AG
~ pro -SSAG.
(4.7.12)
Proof.
Proposition.
If
{G.} 3
by Remarks (4.7.8).
(4.7.13) in
Proof.
pro- SSAG.
All discrete simplicial abelian groups are fibrant.
is discrete, the maps
Gj --->i ~ k
<j
{Gk}
are fibrations
D
Proposition.
pro -SSAG
Prolong the W -construction levelwise to
W
takes a short exact sequence
into a short exact sequence
0---+K--~H-->G--~0
0--->WK--->WG--->WH--->0
Use Propositions (4.6.3) and (4.7.10).
in pro- SSAG.
0
The analogue of Proposition (4.7.11) is difficult to state; the ideas will be used in the latter part of this section.
We shall now use
pro- SSAG
and
Ho(pro -SSAG)
to classify extensions in
pro -AG.
(4.7.14) isomorphism
Proposition.
For
G
and H
in p r o - A G ,
S
induces a natural
157
~
Proof. because
S
Because
o _ AGS(G, H)
S
is full, TS = 1
is full and
~
is a monomorphism.
pro -SSAG,
HOMpr ~
o _ AG(G, H)
To show that
where
G
o
AG(G,H) = HOMpr ~ - SSAG(G,H).
Also,
the induced natural transformation
pro -AG'
E:0 - ~ G
in
>EXtpr o _ SsAGS(G,H) 9
EXtpr o _ SSAG(G, H)
is an epimorphism{ consider an extension
--->X --~ "'" --->X' -->H --> 0
a~d H
are discrete.
Applying
T
to E
yields an
extension
TE:0 --->G --+ TX ---~ "'" --->TX' --->G --->0
in
pro -AG.
Because there is a (natural) map
on
G
E ~ STE ~ Im o ,
and H,
as required.
We shall henceforth simply write
(4.7.15) and for{Q}
Theorem D.
for
which is the identity
[]
Ext
pro - A G
and
Ext
pro- SSAG"
For a levelwise free abelian pro- (abelian group)
in p r o - AG,
ExtS({Gj}, { ~ } )
Proof.
Ext
STE--~E
= Ho(pro-SSAG)({%},
First, consider the case
Ext0({Gj}, { ~ } )
= HOM ({Gj}, { ~ } ) .
s = O,
where
Because
{Gj}
{wS~}).
is free, hence coflbrant
{G S },
158
(see w
w
and [Q- I, w
and
{~)
is discrete, hence flbrant
(Proposition (4.7.12)),
Ho(pro- SSAG)({Gj), {H~}) = [{Cj}, {Hk}] (hamotopy classes of maps with respect to the cocylfnder Because
~)
is discrete,
{~[0,l]} ~ (~).
{Hk}[O'l] - {Hk[O'l]}).
Hence,
Ho(pro-SSAC)({Gj}, {Hk}) = pro-SSAG ({Gj}, {Hk}), as
required. For
s > 0,
we shall use one of B. Mitchell's characterizations of derived
functors [N_it- 1, p. 198, case III]. 0 - - ~ { A i} --~{BI}--~{C i} ---~0 pro -SSAG~ SSAGJ.
Suppose first that
is a short exact sequence of level maps in
that is, a short exact sequence in the appropriate level category Consider a fixed
J
in J.
Then there are fibre sequences
Aj ~ Bj "--~Cj,
and
WAj "-~ ~Bj ~ WCj .
We may obtain a connecting morphlam in
Ho(SSAG),
~:Cj --~ WAj
and thus a
co-Puppe (flbratlon) sequence (each map is the fibre of the next map)
(4.7.16)
as follows. f and g
Aj
~ Bj ~
The homomorphlsms in the diagram
Cj
Bj ~
~WAj
0
and
~WBj ---~WCj
WAj --~ 0
induce the flbrations
159
WA. 3
B.3
l
>WAj x AjBj
f'>WAj Aj x O -= WA.3
O x A.Bj -= Cj 3
Further,
g
i s an e q u i v a l e n c e .
Cj e 0 x AjBj
Then
6
Let
6
[g]-1 ~WAj
be t h e c o m p o s i t e i n
if]
x AjBj
Ho(SSAG)
~WA.3 x A"O3 ~ WAj.
is the required connecting morphism, and it is easy to show that sequence
(4.7.16) is fibration (co- Puppe) sequence.
Therefore, the sequence
(4.7.17)
is an inverse system of long fibration sequences, hence a long fibration sequence in
Ho(pro -SSAG)
by Proposition (3.4.17).
For any
G i n Ho(pro -SSAG),
(4.7.18)
[G,{Ai} ]
(4.7.17) induces a long exact sequence 6,
o f a b e l i a n g r o u p s , where
More generally, if pro -SSAG,
> [G,{Bi}]
[-,-]
[G,{Ci}]
> ~_.[G,~NAi)I
E Ho(pro-SSAG)(-,-).
0--->A--->B--+C--+0
, ...
Compare ( 3 . 4 . 1 6 ) .
is a short exact sequence in
Proposition (4.6.3) and the above techniques yield a long exact
160
sequence
(4.7.19)
[G,A]
analogous to (4.7.18).
9 [G,B]
9 [G,C]
(H,H I) " ([@,-],[@,W'(-)I)
is a connected pair of exact functor on
(H,H I)
pro -SSAG.
If
G
is discrete
and free, and we restrict the functors (4.7.20) to is a connected pair of functors with
pro -AG(G,-). WB = WB i
> ".-
Thus
(4.7.20)
pro -AG)
6 * 9 [G, WA]
Further,
H = Ho(pro -SSA~(G,-)
Ho(pro -SSAG)(G,W(-))
then
=
vanishes on objects of the form
since such objects are levelly contractible
given any short exact sequence
pro - AG,
( in
(see (4.7.9)).
0---~A---~B--->C---~O
in pro- SSAG,
Finally, there is a
diagram
(4.7.21)
O-->A--~
0 ""~ A ~
in which the bottom row is exact. O--->A--~WB--~WB/A--~0 0 ---~A---~B --->C ---~0. that is,
B--~
C
i
WI3 -----~~ / A
--~0
~
0
Thus, sequences of the form
are coflnal in the directed set of all sequences Since
HI(A) = Ext (G,A),
HI(wB) - 0,
H1
is the derived functor of
by M/tchell's criterion.
By iterating the above construction, we obtain the required isomorphisms
ExtS({Gj }, { ~ } )
- Ho(pro - SSAG) ({Gj }, { w S ~ } )
H,
161
for {G.} levelwise free in 3
pro- AG and
The above results hold on
{~}
in pro-AG.
pro- G for s = 0 and i.
We may use the homotopy inverse limit (w
D
Details are omitted.
to reformulate the above theorem
(for (a) and (b) see Bousfleld and Kan [B -K, w
Part (c) is our Theorem F
(w (4.7.22) Theorem. a) Let {Gi} be an inverse system of groups.
Then
f l i m I {Gj} if n ffi0,
w11 (hollmWGi) ffi ~ i i m {Gj}
if nn ffi>I.i'
b) Let {G.} be an inverse system of abelian groups. 3
Then
{wsGj}) = ~0~llmS-n{Gj} if 0 < n < s, wn (holim if
c)
Let
n>
s.
{Gj} be an inverse system of abelian groups, and let
KG. be the simpliclal spectrum obtained from the simplicial 3 prespectrum
{wsGjIs ~ 0}.
Then using stable homotopy groups,
Sn (hollm {KGi}) - If0im-n(G•
n>o.n <__0,
162
Proof.
(a) and (b) follow from the observation
w
(holimWScj)
n
= Ho(pro- SS) (S n, {~Gj})
= Ho(pro-SS)
(S O , {WS-nGj})
= Ho(pro-SSAS)
together with Theorem E. spectra
Sp
w
For (c), use analogous computations with simpliclal
and simplicial abelian group spectra
and consider each
KGj
(Z, {WS-nG.}). 3
SpAG
[Kan- i, 2, 3], or use (b)
as a simpliclal prespectrum.
Strongly Mitta~-Leffler
pro- groups.
We shall give an appropriate generalization
to inverse systems indexed by
uncountable indexin~ sets of the following well-known results.
(4.8.1) surJections.
(4.8.2)
Let Then
{Gn}
be a tower of groups such that the bonding maps are all
limS{Gn } = 0
Suppose further that
for
s > 0.
lim {G n} = 0,
then
A pro -group is said to satisfy the Mitta~-Leffler
{G n} m 0
(M-L)
in pro - G.
condition if it is
pro- isomorphic to an inverse system of groups whose bounding maps are surjections.
Keesling
[Kee -2] has exhibited a M - L pro - (abelian group)
an uncountable directed set, such that
lim {G i} = 0
Thus (4.8.2) fails in general for M - L pro - groups. structed a movable
(Definition
but Keesllng
{Gi},
{G i} ~ 0
indexed by in pro - G.
[Kee- 2] also con-
(4.8.5), below) inverse system of long exact
sequences of abelian groups such that the inverse limit sequence is not exact.
We
18S
shall use this example to prove (Proposition (4.8.6), below) that (4.8.1) also fails, in general, on M -L pro -groups.
In a positive direction, we suggest the following definition as the appropriate generalization
(4.8.3)
of the M - L condition to uncountable inverse systems.
Definition.
Leffler (S - M - L ) that
G.3 ---> l ~ k <
G
J{%}
pro -group
is
j
{Gj}
such
in J,
the
are surjections.
implies M - L .
Proposition
if ~nd only if
is said to satisfy the stron~-Mittla~-
is a cofinite strongly directed set, and for all
Clearly, S - M - L
(4.8.4)
pro -group
condition if it is pro- isomorphic to a
J = {j}
natural maps
A
Also, for towers, M - L
[E - H - 5 ] .
A pro-group
pro -isomorphic
G
to a flasque
implies S - M - L .
is strongly Mittag-Leffler pro -group
indexed by a
is flasque and
J = {j}
cofinite strongly directed set.
We sketch a proof for completeness.
If
{Gj}
cofinite strongly directed set, the natural maps tions by definition.
Conversely,
directed set and the natural maps be an ordered subset of {gj lJ ~ J'} {Gj}
{G.} 3
are surjec-
is indexed by a cofinite strongly
Gj --->lim k <j{Gk}
~re surJections.
Let
J'
It is easy to extend an inverse system
to an inverse system indexed by
is flasque.
(4.8.5)
J.
suppose
Gj --->l~k< j{G k}
is a
J
by induction over
J \ J'
Thus
D
Theorem G.
Let
{Gj}
be a strongly-Mittag-Leffler
pro - group.
Then
164
a)
liml{Gj} ffi0 ;
b)
If
c)
If
(Gj}
llm {Gj} = 0,
Proof.
j
the induced maps
(4.7.11).
{Gj} a 0
J = {J}
the induced map Wp:WGj
W(I~
W
p:Gj --~ l ~ k < j{G k}
is a surjection.
Then,
are fibrations by Proposition
is an adJolnt functor (see, for example, [May - 1, Hence the inverse system
that is, the induced maps
Q'Gi ( in
s > 0;
is a cofinite strongly directed set and
< j {Gk} ) ~ I ~ k < j{WGk}.
is fibrant (w
are fibrations
for
in pro-G.
> W (l~k<j{Gk})
Further, since
Theorem 27.1]), {WGj}
then
We may assume that
that for each
limS{Gj} = 0
is abelian, then
SSG
,- l~bS~< J {WGk}
or SSAG).
By Theorem E (g4.6),
llml{cj } ffi Tfo{W'G j } " "~o(hol~m{W'G}) j (see w
for holim),
Because
{WGj}
is fibrant,
holim {WGj} = lira {WGj}
because
is an adjolnt.
Hence,
(see (4.2.11))
~0(hollm {WGj}) = O.
Part (a) follows.
165
The proof of part (b) uses the formula
limS{Gj} = ~ 0 { ~ G j }
in a similar way.
Details are omitted.
For part (c), assume that for some
j
in J,
Gj
# O.
Choose
g # e
0
(the identity) in
Gj .
For
k ~ Jo
in J,
let
gk
Otherwise use induction on the number of predecessors of
element
{gj}
in lim {Gj}
with
gJ0 = g ~ e.
Hence,
be the image of
j
in J
g.
to define an
lim {Gj} # 0.
This
contradiction establishes part (c).
Recall that an object there exists a
k 9 J
{X.} 3
of pro
- C
is said to be movable if for each > k
such that for all
J
there exists a filler in the
diagram
x~
9
xj
\ \ \ \ k N
It is easy to check that movable
xk
pro- groups
satisfy the Mittag-Leffler
con-
dition.
(4.8.6) Leffler
Pr__oposltlon
pro- groups.
[E -H - 5].
In general,
llm I
need not vanish on Mitta8-
168
We sketch the proof for completeness. pro - groups
{Gj},
If
liml(Gj} = 0
for all
M- L
then any short exact sequence
0 --->{Aj } --~ {Bk} --~ {C~} --->0
of
M -L
pro- groups
would yield a short exact sequence under the functor
In [Kee -2] Keesling constructs a movable pair,
(Xj,Aj)
in
llm.
pro-ANRpairs,
hence a movable system of long exact sequences
~j (4.8.7)
{--.
~HI(Xj) ------>HI(Xj,Aj)
>H0(Aj)
9 ..-} ,
such that the induced sequence
(4.8.8) lim {Hl(Xj)}
is not exact at the middle term.
lim (~j) lim {HI(Xj,Aj)}
> lim {H0(Aj) }
Since the kernels and images in sequence (4.8.7)
are movable, and hence M -L, sequence (4.8.8) would then be exact, contradicting Keesling.
Therefore,
w
lim I
cannot vanish on all
M-L
pro- groups.
The Bousfield -Kan Spectral seqUence In this section we discuss the Bousfield -Fan spectral sequence for the homotopy
groups of the homotopy inverse limit of a pro- (slmplicial set) [B - K, Chapter XI]. Although their model structure [B- K, p. 314]
on SS J
differs from ours, the
167
resulting homotopy categories are isomorphic (see w object in our model structure, then
Let replace
X E SS JX
assume that
Because our
by a fibrant object X
is fibrant.
limit, temporarily denoted
X
X
is a fibrant
~s fibrant in the Bousfield - K a n structure.
holim X'
Also, if
preserves weak equivalences, we may
with
hollm X = holim X'.
Hence, we may
In this case, the Bousfield -Kan homotopy inverse h~
satisfies
Ho(pro -SS)(W,X) m Ho(SSJ)(cW,X) = ~o(SS)(W,
(cW
denotes the constant diagram w~th
Proposition~XI.8.1]. holim X ~ holim B _KX.
cW. = W 3
holi~_KX) for all
J
in J)
[B- K,
By uniqueness of adJoints (see, e.g.~ [Mit-l]), A direct proof also exists, compare Milnor's use of tele-
scopes in [Mil -S].
Hence the following result of Bousfield and Kan holds for our homotopy inverse limit.
(4.9.1)
Theorem
[B-K,
there is a spectral sequence
w
For a pro - simplicial set
w {Er(X),dr} ,
with
E2(X) = {E2P'q(x) ffi limPj {~q(Xj)}},
Under suitable conditions,
{Er(X)}
bidegree
d
r
= (r, r - l ) .
converges completely to
There are two important special cases.
X,
w, holim X.
D
168
(4.9.2)
If
X
in pro - SS,
is equivalent in
pro - Ho(SS,)
to a slmpli-
clal set, then
E~'q(x)
=
E~'q(x) = {~q hollm X, O,
(4.9.3)
If
{Xn, n ffi 0 , i , " ' } ,
X = {Xj) then
in pro -SS, E2(X) = E (X)
p
=
0
p~O
is equivalent to a tower and the spectral sequence collapses to
the short exact sequences
0
~ llmlj{Irq+iXj}
~ q hollm X
; limj{~qXj}
"10.
We shall need the following extensions in order to discuss strong homology (5.6.7) - (5.6.8) and Steenrod homology (w
First, we may replace slmplicial sets by simplicial spectra in the Bousfield Kan spectral sequence.
Their proof of convergence still applies if suitable care
is taken with smash products.
In particular, for
{5}
there is a spectral sequence with
EP'q = limPj{~qS(Xj ^ E)} 2 = limPj{hq(Xj)}
which converges under suitable assumptions to
~ pro - SS,
and
E E Sp,
169 S
~, holim {Xj ^ E)
Sh,{Xj)
(see (5.6.8))
( ~ h, (holim {xj }))
Second, suppose
X = {Xj)
(slmpllcial set) or a
where each
pro - (simplicial
Xj = { X j , k }
spectrum).
is itself
Definition
a pro -
(4.2.10)
implies
that
hollm {Xj, k} = holimj {hollm k {Xj,k}}.
Hence there is a spectral sequence w i t h
(4.9.4)
E~ 'q = limPjwq (hol~ k {Xj,k} )
( ~ llmPj {,q{X~,k>k},
w
see (5.6.1))
Homotopy collmits We shall review the history of homotopy collmits and give a construction dual
to our construction of homotopy limits in w
J. Milnor [Mil -3] introduced the homotopy collmlt of a direct tower in his work on axlomatlzlng additive homology theories on countable
x - {x0 - ~
be a direct tower.
CW
complexes.
-~x 2 -~--.)
Milnor introduced the (direct) telescope of
X
Let
170
Dir tel X = X 0 x [0,i] u X I x [1,2]
u X2 •
[2,3]
--',
shown below
0
I
There is a natural map a tower of cofibrations in
inj -Top,
see w
2
Dir tel X
(i.e.,
X
3
>colim X.
Milnor proved that if
X
is
is cofibrant in the category of direct towers or
then this map is a homotopy equivalence.
Bousfield and Kan used a dual construction
(replacing the infinite mapping
cylinder of a direct tower by the infinite path fibration construction of an (inverse). tower) in their geometric unstable Adams' spectral sequence They extended
this construction
to obtain their
(see [B -K]).
holim.
J. Boardman and R. Vogt's work on homotopy everything H - spaces required the homotopy collmit of a diagram; This construction
an explicit description is given in [Vogt -i].
is dual to the Bousfield -Kan description.
There is also an evident duality between the Bousfield - Kan spectral sequence and the Segal spectral sequence for the classifying
We obtain a homotopy colimit functor
hocolim:
space of a category
Ho(inj -C)
[Seg].
> Ho(C)
171
coadjoint to the natural inclusion
Ho(Top)
~ H o ( i n J -C)
model category ) by dualizing the construction in w is obtained by replacing with
X = X'
in
X
Ho(Inj -C)
by a coflbrant object
(C
a nice closed
Explicitly, X'
in
inj - C
hocolim
X
(see w
and defining
hocolim X ~ colim X'.
Because the colimit functor
collm:
inj - C ---~C
preserves coflbrations and
trivlal coflbratlons (Theorem (3.8.7)),
Ho(C) (hocollmX,Y)
for
X r inJ-C
conclude that for
and X
Y ~ C.
a Ho(inj -C)(X,Y)
Milnor's theorem on telescopes can be extended to
coflbrant in
inj - C 9
from the Vogt homotopy collmit to our hocolim.
ther~ is a natural weak equivalence
w
w
pro - ~.
Introduction.
In w w
THE ALGEBRAIC TOPOLOGY OF
we showed that if
C
is a nice model category then so is
we developed the theory of homotopy inverse limits for
pro -C.
section our main goal is to describe the algebraic topology of compare
Ho(pro -C)
In w
pro - C
In this and to
pro -Ho(C).
we prove comparison theorems which relate maps and isomorphisms in
Ho(tow -C)
w
with
pro -C.
to maps and isomorphisms in
tow -Ho(C).
contains some remarks about completions.
In w
w
we review the Artin-Mazur theory of
pro - Ho(SS,).
is concerned with Whitehead and stability theorems and counter-examples.
In w
we introduce strong homotopy and homology theories and prove a Brown
theorem for cohomology theories.
w
Ho(tow-C,)
Let
C,
tow-Ho(C,).
be a pointed nice simplicial closed model category which satisfies
Condition N of w functor.
versus
and let
~:Ho(tow -C,)
~tow -Ho(C,)
denote the natural
In
173
(5.2.1)
Comparlson Theorem.
There is a natural short exact sequence of
pointed sets:
(5.2.2)
0 ---~liml k colimj {Ho(C,)(E X~,Yk)}
~Ho (tow -c.) ({xj),{Yk}) 9 tow - Ho (C,) ({Xj }, {Yk ))
(5.2.3) Remarks. Ho(tow -C) ({Xj},{Yk})
~0.
It is easy to see that the natural map , tow -Ho(C) ({Xj },{yk} )
is always surJective; we need base
points only to make a more precise statement.
(5.2.4)
Remarks.
J. Grossman [Gros -2] obtained the above sequence in his
coarser homotopy theory of
tow- SS,.
Proof of Theorem (5.2.1).
By Axiom M2 for
C, N,
ment, we,.can replace any tower by an equivalent (in i.e., tower of flbrations of flbrant objects. is fibrmnt.
Define
Y-I = * ;
or an easy inductive arguHo(tow -C,))
fibrant tower,
We may therefore assume that
then the map
Y0---~Y 1
is a flbration.
Consider the following function spaces (all of which take values in w
HOM (Xj,Yk) , J ~ O,
(5.2.5)
k ~-i;
HOM ((Xj},Yk) -- colimj (HOM (Xj,Yk)},
HOM ({Xj},(YkD m i ~
{Yk }
k >__0;
(~OM ({Xj),Yk)},
SS,;
see
174
The fibrations
Yk--->Yk_l
the simplicial structure on HOM ({Xj},Yk)
induce fibrations C,,
HOM (X.3,Yk)
> HOM (Xj,Yk_I)
by
and also fibrations
9 '>HOM ({Xj},Yk_l)
by the simplicial structure on
pro-C,
or a
direct argument (the colimit of a sequence of Kan fibrations is a Kan fibration by the "small-object argument" [Q-i, w
Because (5.2.5) expresses
H0M ({Xj},{Yk})
as the limit of a tower of
fibrations, the Bousfield-Kan spectral sequence for a tower ((4.9.3)) yields an exact sequence of pointed sets
(5.2.6)
0
> limlk{~l(HOM ({Xj},Yk))}
~ ~0(HOM ({5},{Yk}))
---> I ~ { ~0 (HOM ({Xj },Yk)) }
>0.
By the above simplicial structures, ~I(HOM ({Xj},Yk)) = [S I, HOM ({5},Yk)] -~ ~0(HOM (SI, HOM ({Xj},Yk)))
-= ~0(HOM ((Xj} ^ SI,Yk))
-_-~0(HOM ({Xj
~ SI},Yk))
a ~0(HOM ({ ZXj},yk) )
-= Ho(pro -C,) ({ 7.Xj},yk)
colimj [ Z Xj,Yk ]
175
([-,-]
denotes
Ho(C,)(-,-);
extension property).
the last isomorphism follows by the homotopy
Also,
9o(~OM ({Xo }, {Yk }) = Ho (tow - C,) ({Xj }, {Yk}),
and
.o(HOM ({Xj},Yk)) = colim j {[Xj,Yk]} ,
as above.
Because the above isomorphisms are natural (w
tow -Ho(C,) ({Xj},{Yk}) -: limk{colim j { [Xj ,Yk] } }, (5.2.6).
(5.2.7)
and
the conclusion follows from
D
Remarks.
In the above proof, one can replace the Bousfield - Kan "lim I argument"
spectral sequence by a direct argument dual to Milnor's
[Mil -3,
Lemma i]; see, for example [B- K, p. 254].
Theorem (5.2.1) suggests the following.
(5.2.8)
Question.
tow -Ho(C), ~f,
Let
f
is invertible.
be a map in Is
f
Ho(tow -C)
invertible in
whose image in Ho(tow -C)?
This question appears quite difficult, and its analogue in proper homotopy theory (see w
has attracted recent interest (Chapmann and Siebenmann [C -S]).
At present we can only offer a partial answer (Theorem (5.2.9)). (5.2.9) in proper homotopy theory [E -H -3], (see w [C = S ] .
The analogue of
answers another question in
176
(5.2.9) image
~f
Theorem. in
tow -Ho(C)
g:{Xj} --->{Yk }
Proof. {Xj}
and
both
~f
Let
in
f:(Xj}
) (Yk }
is Invertlble.
Ho(tow-C)
with
be a map in
Ho(tow -C)
whose
Then there is an isomorphism
~f ffi~g
in
tow-Ho(C).
As in the proof of Theorem (5.2.1), we may assume that both {Yk }
are fibrant.
and its inverse in
diagram over
By relndexing if necessary, we may then realize tow "Ho(C)
in the following homotopy-eommutatlve
C.
Xk+l
) Yk+l
(5.2.10)
> Yk
1 Let
Z
I
be the tower
hl X0<
1
fl Y1 <
fk
~
hk+l
Yk+1 '
fk+l
~+i"
hk+l
""~ 9
177
Form the homotopy c o m m u t a t l v e d i a g r a m
(5.z.iD X j_ '''~-
'~
bond
(
Xk+ 1
id
hk+l Zi " ' ' < -
~.-,,
fk+l Yk+l ~
hk4-2 Xk+l . . . .
Yk+2 <
" ""
Yk+2 ~
"" "
bo t-~l Yk+l ~
Diagram ( 5 . 2 . 1 1 ) Id'~
--~
and
factors
~f
i d : Y k ---~Yk
in
tow-Ho(C).
By i n d u c t i v e l y
a b o v e , we may o b t a i n a s t r l c t l y
d e f o r m i n g t h e maps commutative diagram
178
(5.2.12)
hood ~+I
i
l=
i
7. "'" 9
f +l
X~<
~-
Yk+l ~
~+i
Yk+l <
posite map
"""
u
"""
bond
y ....
such that diagrams
Yk+2 <
<
(5.2.11) and (5.2.12) are equivalent over
g:X--->Y --->Z
in diagram
(5.2.12)
Ho(C).
is invertlble
in
because it is the composite of two levelwlse weak equivalences, ~g
=
~f,
as required.
(5.2.13) isomorphic
in
isomorphism
Proof. in
Corollary.
in
If either
tow-He(C)
X
or
to an object of
Y
and satisfies
is stable in Ho(C)),
then
tow -Ho(C) f
(i.e.,
above is an
Ho(tow -C).
say isomorphic
of the theorem shows that to objects
X'
and Y'
Consider the composite map
(5.2.14)
Ho(tow - C)
0
A first application
Ho(tow -C),
The com-
X'
--" X
f
Y
-= Y' .
X of C,
and Y
are stable
respectively.
179
For any
Z
in C,
Ho(C)(Z,X') = tow -Ho(C)(Z,X') tow -Ho(C)(Z,Y') Ho(C)(Z,Y').
Thus the composite (5.2.14) is an isomorphism in Ho(tow -C).
f
Remarks.
If
Ho(C,)
is an isomorphism in
We uowmake
(5.2.16)
Ri~Idification.
{Xj}
is abellan, t h e Comparison Theorem implies
Ho(tow -C,).
the following observation.
image of an object in replace
hence in
0
(5.2.15) that
Ho(C),
Each object in
Ho(tow -C).
tow -Ho(C)
To see this, given
by a tower of flbrant objects
{X'j},
is equivalent to the {Xj}
in tow -Ho(C),
and choose representatives
for the bonding maps of the latter tower.
(5.2.17) tow -Ho(C)
Proof.
Corollary. and
The isomorphism classification is the same in
Ho(tow-C).
Use (5.2.9) and (5.2.16).
D
The above results give a usable relationship between the weak and strong homotopy theories of towers,
See w
especially w
proper homotopy theory of o -compact spaces, and w the shape theory of compact metric spaces.
where towers are used in the for a similar application to
180
The Comparison Theorem (5.2.1) is replaced by
Off towers, little is known.
the following extension of the Bousfield -Kan spectral sequence (see w
(5.2.16)
For
Theorem.
{Xj}
and
{Yk}
in pro -C,,
there is a spectral
sequence with
E2p,q = limP k {colimj { [ z q x j , Y k]}},
which is closely related to
Proof.
We may assu|
Ho(pro -C,)({Xj},{Yk} ).
that
{Yk}
is flbrant.
{J}
Because
is filterlng,
collmj {[Z qS'Yk] } " Ho(pro-C,)({ Eqxj},Yk)
(each map on the left may be represented by a map By the slmplieial closed model
structure
on
Z qxj ---~Yk
pro - C,,
Ho(pro-C,)({T. qxj}.Yk) ~ ITqCHOM({Xj}.Yk)). and
Ho(pro -C,)({Xj}.{Yk} ) ~ ~o(HOM ({5}.{yk})). where
HOM
is the "function space" of w
HOM ( ( 5 } , { Y k } )
Finally,
= i ~ k {HOM ({Xj},Yk)}
(essentially by "enriched adJunctlon," w
= h o l ~ k HOM ({Xj},Yk) (because
holim ~ lim
flbrant objects).
on
f o r some
j).
181
Applying the Bousfield -Kan spectral sequence to desired result.
HOM ({Xj},Yk)
gives the
D
Unfortunately, we cannot conclude that ~:Ho(pro -C,)({Xj},[Yk} )
9 pro -Ho(C,)({Xj},(Yk})
The ri~idification question - which objects of Ho(pro -C) -
pro -Ho(C)
is unanswered and appears quite hard.
question about slmplicial objects over
Ho(SS)
is onto.
"come from"
Alex Heller asked a similar
and over
SS.
isomorphism classification question (compare (5.2.9) - (5.2.15))
Also, the is unanswered
off towers.
Thus, our present knowledge suffices for the proper homotopy theory of o -compact spaces (see w
and the shape theory of compact metric spaces (see w
but not for more general spaces.
Similarly, there are good comparison theorems relating the strong and weak homotopy theories of direct towers. Ho(inj -C)
w
and
inj -Ho(C)
Also, similarly, the relation between
remains obscure.
Remarks on Completions. In 1965, Artin and Mazur [A -M, Chapter 3] introduced the profinite completion e pro -Ho(CWo)
of an object
theorems in ~tale homotopy theory.
X ~ Ho(CWo)
in order to prove comparison
The inverse system
X
category which has as objects based homotopy classes of maps X
is indexed by the X--->X
where each
has finite homotopy groups, and has as morphisms homotopy commutative triangles
182 X
/'\ Then the association
(X--+X)
---->X=
yields an inverse system
Sullivan's
X.
G/PL
work on the Adams conjecture and the homotopy type of spaces such as to study the functor ing the funetors
lim~{[-,X ]}
[-,~],
[Sul -i],
[Sul - 2].
led him
By suitably topologiz-
Sullivan showed that the functor
lim {[-,X ]}
satisfies the Mayer - Vietoris (exactness) axiom as well as the wedge axiom, and hence
lim { [-,X ]}
is representable by Brown's Theorem [Bro].
concentrated on the complex
X
which represented
Sullivan then
llm [-,X ],
[-,~] ~ Lim {[-,~ ]);
(5.3.1)
being a simpler and more familiar object than
In 1972, Bousfield and
{X }.
Kan [B - K], motivated by their work on the Adams spectral sequence, defined for every commutative ring R X.
They obtain
{R X}. s
R
R X
and pointed simplicial set
X
a functorial R - completio%
as the simplieial inverse limit of a tower of fibrations
In this situation it is n_oolonger true that the funetors
lim s {[-,RsX ] }
are naturally equivalent.
of pointed sets [B-K]
(5.3.2)
[-,RX]
and
Instead, one has a short exact sequence
(see w
0 --+ iimls{ [ E W,RsX]} --* [W,R=X] --->liras{ [W,RsX]} --*0.
We shall briefly compare (5.3.1) and (5.3.2), modulo rigidification problems (see the end of w
In (5.3.2), we always have
183
[W,R X] e Ho(pro-SS,)(W,{RsX)) this sense,
because
R X = holims{RsX}
term of (5.3.2).
If
In
because
[-,R X] # lims{[-,RsX]}
and this difference is measured by the
Ho(pro -SS,)(-,{RsX}) # lims{[-,RsX]} , llm I
(see w
R = Z,
and
X
is a simply-connected (or even nil-
potent [B -K, Chapter 3]) finite complex, then [B- K]
{Z X}
is cofinal in
S
{Xa}.
Also, for
lim 1 {[ ~ W,ZsX]} S
(5.3.3)
W
finite, the groups
vanishes.
{[ E W,ZsX]}
are finite, so
This s u g g e s t s the f o l l o w i n g .
Proposition.
{X } be a tower of pointed, connected
Let
(SS or
n
CW)
complexes.
If
is representable and
limn{[-,Xn]}
limln{[ E -,Xn]}
vanishes, then
[ - , h o l i m {Xn }] i limn { [ - ' X n ] }
on all pointed complexes.
Proof.
Let
Q
represent
limn{[-,Xn]}.
[-,Q]
~ lira n { [ - , x n]}
\
(5.3.4)
Consider the diagram
x \
/ \ \
\ [ - , h o l i m {Xn}]
Evaluation of the top row on
holim {X }
yields the filler which makes (5.3.4)
n
commute.
Vanishing of
~i(holim {Xn})
limln{[ E -,Xn]}
> limn{~i(Xn)} ,
i ~ 1,
implies that the group homomorphisms are isomorphisms (see w
also
184
(5.3.2)). w i ( h o lml is
Then diagram (5.3.4) yields isomorphisms {Xn} ) e wi(Q),
connected,
holim {X } n
the Whitehead Theorem.
w
i ~ i. and
Because vanishing of Q
lim I
implies
holim {Xn}
have the same weak (singular) homotopy type by
The conclusion follows.
D
Some basic functors. Artin and Mazur [A -M.
of an object in
w167
pro -Ho(SS0) ,
introduced the homology and homotopy pro - groups as well as Postnikov decompositions,
Theorem, and a type of Whitehead Theorem.
the Hurewicz
In this section we shall review the
above results, except for the Whitehead Theorem.
The Whitehead Theorem will be
discussed in w
Recall that any covariant functor pro -T:pro -C
", pro -D.
pro -homology functors on
T:C--->D
We may therefore define the pro - homotopy and pro -Ho(SS,)
(5.4.1)
pro
A
is an abelian group.
homology theory. on
pro -SS,
structure. on
by the formulas
-.i({xj)) ~ {.i(xj)},
pro-Hi({Xj};A)
where
may be prolonged to a functor
m {Hi(Xj;A)}
A similar formula holds for any generalized
These functors induce pro - homotopy and pro - homology functors which satisfy the usual properties with respect to the closed model
(Note that fibre sequences and related constructions are no___~tfunctorlal
Ho(SS,),
pro -Ho($S,).)
hence it is difficult to describe homotopy and homology theories on Artin and Mazur even define homology with twisted coefficients; we
185
shall not need these formulas in our work.
Because cohomology is contravariant, the analogue of formula (5.4.1) for cohomology takes values in the category groups.
Because
colim: inj -AG
inj - A G
> AG
of direct systems of abelian
is exact, Artin and Mazur define the
cohomology groups by
fii({xj};A) ~ colim.3 {Hi(Xj ;A)}"
(5.4.2)
The category
K0
of pointed, connected Kan complexes (= fibrant simpllcial
sets) admits functorlal Postnikov -type resolutions (see, e.g., [May -i], or [A -M;w
We shall describe these resolutions and the induced Postnikov - type
resolutions on
pro-Ho(SS0).
Let
An
denote the p-skeleton of the standard P
simplieial n - simplex
An.
By analogy with the formula
set of n -simpllees of a simplicial set
X,
for
X
in K 0
X
= Ss(An,x)
n
let
for the
cosk X P
be
the simpllcial set whose n - simplices are given by
(coskpX)n = SS(Anp,X),
together with face and degeneracy maps induced by the coface and codegeneracy maps d i :An-I
) An P
and
is obtained from to maps from
si:A n+l
P
>A n P
X
Anp
compatible maps are not fibrations.
for
0 ~ p ~ n.
by adjoining additional n -cells for to X. coskpX
The inclusions >
Roughly,
P
coskp_iX
and
A np --->Anp+l X
> coskpX.
n > p and
cosk p X P
corresponding Anp --->An
Caution:
induce these maps
We may define the coskeleta of an arbitrary simplicial set
X
186
by the formulas
cosk Ex~X P
weakly equivalent to
X).
(recall that For
X
Ex X let
in K 0,
is a Kan complex naturally X (p)
and X(p)
be the
homotopy -theoretic fibres of the maps below:
X (p)
> coskpX,
>X
(5.4.3) X(p)
Because
SS 0
admits canonical factorizations of maps as trivial cofibrations fol-
lowed by fibrations ( [ Q - I , in
X.
> cosk X. P
~ coskp+iX
Further,
w
see w
cosk X P
MacLane space of type
the sequences (5.4.3) are functorial
is (p -i) - connected, and
K(~p(X),p).
X.
In fact, the above constructions are functorial on
(i)
(ii)
X
Ho(SS0);
the
pth
co-
is characterized by the properties:
~i (coskpX) = 0
for
The canonical map
i ~ p;
X---> cosk X P
to maps into objects
Similarly, the fibre X (p) --->X ---> cosk X P properties.
is an Eilenberg -
We therefore regard the sequences (5.4.3) as
the canonical Postnikov resolution of
skeleton of
X(p)
X "p"
Y
is
with
is universal with respect ~.(Y) = 0 i
(p - i)
is trivial, and the map
for
i ~ p.
connected, the composition X (p) --+X
is universal for these
187
Following Artin and Mazur, we define the POstnikov system of an inverse system X = {Xj}
in either
p r o - SS 0
or
(5.4.4)
to be the inverse system
X # = {coskpXj}
indexed by
{(p,j)}.
pro -SS 0
pro -Ho(SS0)
and
Clearly
pro -Ho(SS0)
to
@
extends to functors from pro -Ho(SS0).
(5.4.5)
If
{Xn}
pro - SS 0
is a tower, then
X ~ a {cosknXn} ,
so we may restrict (5.4.4) to a functor from towers to towers. f:X-->Y
to
in
pro -Ho(SS0)
(respectively,
Ho(pro -SS0))
A map is called a
-isomorphism if it induces an isomorphism on Postnikov systems
f#:X#--~Y ~ .
By using the above machinery and a spectral sequence argument, Artln and Mazur proved the following.
(5.4.6) i < n,
Hurewicz Theorem for
where
n
is an integer
pro -Ho(SS0). > i.
Let
pro -~i(X) = 0
for
Then the canonical map
pro -Wn(X)
) pro -Hn(X)
is an isomorphism of pro -groups.
w
Whitehead and Stability Theorems. In this section let
SSAG,
Sp
Then the Whitehead Theorem (5.5.1) holds in the category
C0
objects in
C.
C
be any of
SS,
SSG,
(simplieial spectra). of pointed, connected
188
(5.5.1)
Whitehead TheQrem in
induces isomorphisms
f,:~i(X)
Ho(Co). > wi(Y )
A map for
f:X--->Y
i ~ 1
in
Ho(C 0)
which
is an isomorphism in
Ho(C0). A natural question is whether (5.5.1) can be extended to Ho(pro -C O )
if the homotopy groups
homotopy pro -groups
{~i(Xj)}
when is an object of
pro -Ho(C0)
~o(C0))
~i(X)
(X E CO)
({Xj} E pro - CO). or
pro -Ho(C 0)
and
are replaced by the The stability problem (i.e.,
Ho(pro -C O )
isomorphic to an object of
will also be studied using homotopy pro - groups.
The following example
shows that additional hypotheses are needed for a Whitehead Theorem in pro -homotopy.
(5.5.2) all
Example.
i ~ i,
because
hut
Let
S~
for infinitely many
there is an essential map
f:X--+Y for in
i ~i.
pro -Ho(C0) Then
pro -Ho(C0).
isomorphism in
f
i,
(see e.g.,
pro- ~i(S ) = 0 pro -Ho(SS0).
for In fact,
[Spa, Corollary 9.7.6]),
S ~ --->S 3 .
Whitehead ~heorem in
in
Then
is not equivalent to a point in
w.(S 3) # 0 i
(5.5.3)
S~ s {V i > n Si }n> 0
pro-Ho(C0).
induces isomorphisms induces an isomorphism
Suppose that a map f,:pro -~i(X) f~:X+--->Y #
> pro -~i(g) of Postnikov systems
Under either of the following additional conditions, pro - HO(Co):
f
is an
189
(a)
supj,k{dim (Xj), dim (Yk)} < ~;
(b)
For each
J,
dim ( 5 ) < |
dim (Yk) < ~,
Proof.
For
C = SS,
Corollary (4.4).].
f
k,
is movable.
the first part is due to Artln and Mazur [A-M,
Their proof uses a spectral sequence argument and easily
ex~ends to the other
Similarly, for
and
for each
C.
C = SS,
the second part is due to the first author and
Ceoghegan [E -G-1,211 see also [A-M~ Theorem (12.5)], [Mos -i], [Mar -i], [Mor ~I]. that
f
(a), let such
chaL
By relndexing as in w
We shall sketch the preef. is a level map
we may assume
indexed by a directed set.
{fj:Xj --->Y.}j
n ffisupj{dim (5), dim (Yj)}.
Consider a fixed
j.
For case Choose
Jl
the diagram
>~)(Yjl )
~l(Xil )
/ / / / / / / / /
1
> ~l(Yj)
~l(Xi ) admits ~ filler.
Then tile map
Convert
fj
and f. Jl
' X~l ' ----,=l(Y~,~) Wl(YJI'
into cofibrations
f'. J
is
0.
and
f
'
Jk
~en the composlte~p
.
190
y~ i ~---->y '. Jl 31
(Y~i IJ
X~ Jl
denotes the i - s k e l e t o n
into X~ 2"
of
Y~ ) i
similarly for all
k
-~y,. 3
may be "deformed" relative tO
with
k > Jl" =
Now choose
" J3 ~ J2'" 9"'in = > 3n-i
so that similar results hold for
conclude that the map
y! -->Y~ 3n 3
(because
dim (Yjn) ~ n ) .
a homotopy inverse to
(5.5.4)
Lemma.
f
~2,~3,
can be deformed relative to
This argument is due to Marde~id
"'"'~n" X~ 3n
Let
be inverse systems in a category
X = {Xl,pll,,A}
C
We
into X~
[Mar -i] and yields
in case (a) by using the following l ~ a
[Mor-l].
J2 > Jl" =
and
over the same directed set
of K. Morlta.
Y = {Yl,qll,,A}
A ,
and let
f% {X 1
> YI}
be a level morphism in
pro -C
iff for any
exists
~l :Y~
~X l
I 9 A
pro - C .
there is some
for which
~I~ f
Then
p 9 A
= Plp
and
f
is an isomorphism in
such that
fl~l~ = ql~;
filler exists in the following solid arrow commutative diagram
f X
P '.J
lP>'l~r
XA
",Y /
/ :~'I I.~/... / /
I~
// Iqx1~ ~ Y1
i ! p
and there
i.e., a
191
Case (a) follows.
In case (b), consider a fixed ~ k,
j.
There exists a
k ~ j
such that for all
homotopy fillers exist in the diagram
f
ond
\bond
Xj
~
Now use the above argument with map
Yk--+Yj
into
An object
X E pro -C
to an object of
C.
which imply that
X
X 9 pro -C O If
y. 3
n = dim ( Y k , ~ )
relative to
~.
As above, the conclusion follows.
is said to be stable if it is isomorphic in
is stable.
If
X e pro-Ho(C0)
and let
an isomorphism if
pro - C
h:holim (X) --->X
is stable for all X
X
is stable, then so are its
pro -~i(X).
Th__eeStability Th.eorem i__n_n pro - HO(Co)
pro -~i(X)
to obtain a deformation of the
The stability ~roblem is the problem of giving criteria on
homotopy pro - groups
(5.5.5)
X.3
bond
[E - G - i].
be the canonical map in
i ~ I,
then
h~
Let pro - HO(Co).
is an isomorphism,
satisfies either of the following conditions:
h
is
192
(a)
supj{dlm (Xj)} < ~;
(b)
X
is dominated in
We sketch the proof. for w
s > 0 h
(w
pro -Ho(C0)
Because
pro
by an object of
-
~i(X)
is stable,
limS{~i(Xj) } = 0
By the Bousfield - K a n spectral sequence (see
Theorem C).
induces isomorphisms
~i(holim (X)) ~ pro -=I (holim (X))
Therefore
Ho(C0).
h#
pro -~i(X).
is an isomorphism by Theorem (5.5.3).
To show that
h
is an isomorphism in case (a), the first author and Geoghegan
applied Wall's finlte-dlmensionallty criterion [Wall] to the homology and cohomology groups of
holim (X)
(which are isomorphic to the homology pro - groups and
cohomology groups of the finite-dimenslonal system phism by [A -M]).
Th~s shows that
dlmensional slmpliclal set.
In case (b), let Y
in C O ,
and
homotopy idempotent
because
h~
is an isomor-
has the homotopy type of a finite-
The conclusion then follows by Theorem (5.5.3).
u:X-->Y
du = i x.
hollm (X)
X
and
d:Y--->X
be the domination maps with
One then applies Brown's Theorem [Bro] to split the
ud:Y-->y
((ud) 2 = u(du)d = ud)
through
Z.
One easily
checks that
X-" { Y <
Y<
Y<
"''} = Z.
Then the ordinary Whitehead Theorem implies that the composite map is an isomorphism.
The conclusion follows.
holim X - - + Z
193
(5.5.5.a)
Remarks.
assumed to be in
The argument given in part (b) shows that if
pro -Ho(C0) ,
it still follows that
X
is stable.
X
is only Dydak has
recently shown that the same conclusion holds in part (a).
So far, we have only been able to prove the following strong tower versions of Theorems (5.5.3) and (5.5.5).
(5.5.6)
The Whitehead Theorem i__n_n Ho(tow -Co).
induces isomorphisms
f~:pro -~i(X)
induces an isomorphism
f#:X#--->Y #
phism in
if
Ho(tow -Co)
f
~ pro -~i(Y) in
Suppose
f:X-->Y
in tow - C O
for all
i ~ i.
Then
Ho(tow -Co).
f
f
is itself an isomor-
satisfies either of the following additional condi-
[ions:
a)
sup {dim (Xj),
b)
f
dim (Yk)} < =;
is movable.
We sketch the proof. Grossman [Gros -2].
For
C = SS
For other
C,
the first part of this theorem is due to
the proof is similar and omitted.
of the second part follows from w
The proof
and appropriate filtered Whitehead theorems,
which are proved in an identical manner to the proper Whitehead Theorem occurring in [Br].
(5.5.7)
The Stability Theorem in
h:holim (X) --->X
Ho(tow-C0).
be the canonical map in
stable for all
i ~ i,
then
isomorphism in
Ho(tow -Co)
h# if
Let
Ho(tow - CO).
is an isomorphism in X
X ~ tow-C O If
and let
pro - ~i(X)
Ho(tow -CO).
h
is is an
satisfies either of the following conditions:
194
a)
sup (dim (X.)} < ~; 3
b)
X
is dominated in
Proof.
pro -Ho(C 0)
by an object of
The first part follows from the first part of (5.5.6).
second part, Theorem (5.5.5) implies that an object Q
Q
of Ho(C0).
in Ho(tow -Co).
an isomorphism in
(5.5.8) if
h
Ho(tow - CO).
Remarks.
canonical map
h:holim (X) --->X
X e pro -Ho(C)
is dominated by
in pro -Ho(C)
tow -Ho(C)
by P.
Y E tow -C dominated by
with
holim,
X
to
is isomorphic to
now implies that
h
is
by an object in
and hence shows that if by an object of
is an isomorphism in P E Ho(C)
Y
Ho(C),
Ho(pro - C).
then the If
(i.e., we are given m~ps
in pro -Ho(C)),
...}.
Ho(SS0),
In fact, Porter's argument depends
Ho(pro -C)
du = i X
y = {p
then
X
is isomorphic in
is easily seen to be dominated in
Recent work of Chapman and Ferry [C -F] seems to imply that if
is dominated in P
Ho(pro -SS0)
Ho(pro -SS0).
is dominated in
to
tow-Ho(C0)
D
is dominated in
is an isomorphism in
pro -Ho(C)
holim
For the
Porter [Por -2] has given a simple argument which shows that
X ~ Ho(pro -C)
d ~X u
is isomorphic in
Theorem (5.2.9) then implies that
only on the functorial properties of
P <
X
The properties of the functor
X E Ho(pro -SS0)
then
Ho(C0).
tow -Ho(C)
in Ho(tow -C),
will also be stable in
by a
P ~ Ho(C),
then
Y
and hence stable by Porter's argument.
pro -Ho(C).
Thus, an object of
if and only if it is dominated by a stable object.
pro -Ho(C)
is also Thus, is stable
X
195
A map
f
f:P --+ P
is said to be a homqtopy idempotent
is said to split through
Q
if there exists maps
P <
if d>
f2 = f
Q
in Ho(C).
such that
u
du = IQ
splits.
and
ud = f
in Ho(C).
The above shows that every homotopy idempotent
This may be used to show that every homotopy idempotent on an s
is homotopic
to a strict idempotent and every homotopy Idempotent
Q -manifold
M
is homotopic
to a strict idempotent
f
-manifold
on a compact
if and only if a certain Wall
+ obstruction
W(f) ~ KO(~I(M))
The development Theorems
vanishes.
of coherent prohomotopy
(5.5.6) and (5.5.7)
to
theory should enable one to extend
Ho(pro -Co)
(see [Pot -3]).
Below we present a number of examples which show the precision of the above results.
(5.5.9)
Ex___ample.
Example
(5.5.2) showed the need of some extra condition,
such as conditions a) or b) of (5.5.3), phism.
in order that a # -isomorphlsm
In this example we show that it is insufficient
be movable.
to require
be an isomor ~ X
and Y
Such an example was first constructed by Draper and Keesling in
[D -El.
Let
S
n
=k/
i>n
Si
and
i :S ---> n n Sn-i
be the natural inclusion.
+These observations were obtained in a conversation with T. Chapman.
Let
to
196
Xn = H k < n S k
= Yn"
Let
Pn:Xn~Xn_l
,
bn:Yn ~ Y n _ l
,
and
fn n:X ~ Y n
be
defined by the following diagram
1 x ... x i x 0. S ] x --- x Sn_ I x S . n
S I x --- x Sn_ I x Sn
i ... S1
Then
X = {Xn,Pn}
, . .
and
Ix --
Sn_l
Y = {Yn,bn}
# -isomorphism (easy), but
f
(5.5.10)
~SI • - -
are movable and
S~--->X
x Sn_ I
f = {fn }
is not an isomorphism in
there is an obvious essential map S~--->xf-~Y
, ...
.-- x O
is a
pro - Ho(CW0);
in fact,
such that the composition
is inessential.
Examples.
We will show in the examples below that the phenomena
exhibited by examples (5.5.2) and (5.5.9) can he realized by inverse systems of finite complexes. J. F. Adams.
These are much deeper examples, though all the depth comes from
Recall Adams' essential map of Moore spaces
finite complexes of the form
Sk o
D k+l)
q
Y
(these are pointed
[Adams - 2]
A: [ 2ry -->y ;
this map is detected by the isomorphism
* ~ A :K(Y)
where
K
denotes reduced K - theory.
=
'K(I
2ry)
# 0,
Hence, the composite maps
197
A
are all of
A
essential.
the inverse system
Thus
is not equivalent to a point in
for all S~
1 2 r m - 2r A : 1 2 r m
.....
i > i.
y
.> [2rny
Z ~ { I 2rny}
pro -Ho(CWo)
bonded by suspensions
even though
Applying the construction of Example (5.5.11) to
pro- ~i(Z) = 0 Z
in place of
yields a map between movable towers of finite complexes which is a
# -isomorphism but not an isomorphism in
pro -Ho(CW0).
The following example provides counter-examples
to many conjectures
(see
[E -H -4]).
Let n > i, n- 1
be the inverse system with
and with bonding maps S 2r
Let
Fn,
{Bn }
Bn
.--->
B0
n 2r B n = Hi=IS
a point, and
for
given by projection onto the last
Bn_ 1
factors.
E n = Y x Bn,
the fibre of
fn:En--->En_l
yet no composite
Pn'
and let
is
Y.
Pn:En = y x B n
> En-k
be the projection;
thus
We shall define "twisted" bonding maps
so that the restrictions
En
~B n
fn Fn:Fn
factors through a
Form the commutative solid-arrow diagrams
~Fn-l
Bn_ I.
are null-homotopic,
198
Ef n
En
y x (S2r~ n
On > y x (s2r) n
(s2r) n
(E'
En-i
,
(s2r) n
>y x (s2r)n-1
....
>
[S2r) n-I
II
{t
(l
Bn
Bn
Bn_ I
p'
is the pullback, and n
is the projection).
Define the filler
n
requiring
P t non - Pn'
that
On
and letting the composite mapping
y x (s2r) n - [ ~ u
x (s2r) n
>y
be the composite mapping
y
here
Wn
•
(s2r) n
id•
n
y • s2r '
is ~he projection onto the first
Finally, let
bonding map
the
fn:En
> y A S2r
$2 r
> En-i
A
~ y;
factor.
be given by the composite
On En
{Fn
~ E'n
9 En-l"
This yields the tower
{En}
and
tower
of flbratlons
Pn > En------>Bn}.
Claim i.
fn[Fn:F n ~
The tower
Fn_ 1
F
is contractible; since the bonding maps
a r e g i v e n by the c o m p o s i t e s
by
189
F
= y
, y • , .. > y • S 2r -..>y A S 2r
A>y
,
n
{F }
is isomorphic in
Pro-Top
to * .
We may use the basepoints in the
{Sn:B
n
>En}.
Since
Fn( = Y)
{pn}{S n} ffi id _
{Bn}
,
to define a section
to show =hat
{pn }
is not inverti-
ble, it suffices to verify the following.
Claim 2.
arbitrarily
{Sn}{Pn} # id 9 {E n}
large
n
in Pro - Ho(Top).
and s u i t a b l e
m
Ass,-ne otherwise, then for
(depending upon
n,
but with
n-m ~0)
the diagram
E
=
y • (s2r)n
Pa
~
(S2r)~
sn
>y
•
(s2r) n
n
y x (s2r) n-m
Pn-m> (s2r)n-m
bond
/
/
E
n-m
would commute up to homotopy.
s
/
x (s2r) n-m
200 Consider the subdiagram
y x (s2r)n
bond > y x (s2r)n-m
l
Pn-m
(s2r) n-m
bond
> y x (~2r)n~m
y x is2r)n-m
Note that all of the above maps are products with
id(s2r)n_m.
fact, and projecting the lower right corner onto
Y,
Hence, by this
we obtain a homotopy commuta-
tive diagram
Y • (s2r) m =
>Y
I g
i =
Y
where
g
is induced from the bonding maps.
Y • (sZr) TM -
Since g*:K(Y)
K(YA
(s2r)m)
~ K(y x (s2r)m)
9 ~A
I
;y
By construction,
(s2r) m = X 2~5~
is a direct summand in
g
Am~ Y 9
K(y x (s2r)m),
is non-zero, hence Claim 2 holds.
Q
is the composite
201 Let
~, ~
(5.5.10a)
~ w. ~ [ ~ si,-]. i= I i i=l
Proposition.
pro - ~,{pn}:{~,(En)}
9 {~,(Bn)}
is a p r o -
isomorphism.
Proof.
This follows from chasing in the commutative solid arrow diagram
s ~,(F n)
) ~,(E n)
01
n*
~ -pn,_ ~ ~,(B n)
i
J Pn-l*>
~,(Fn_ I)
(5.5.10b)
> g,(En_ I)
Proposition.
~ ~*-i (Fn)
1 ~,(Bn_ I)
0 ) > ~,_l(Fn_l) 9
An infinite dimensional Whitehead Theorem fails in
shape theory.
Proof. metric
Let
spaces
E = lim En,
B = lim Bn;
then
p:E -->B
is a map of compact
which is an isomorphism on Cech pro- homotopy groups
(pro-~,)
but not a shape equivalence.
(5.5.10c)
Remarks.
p
(5.5.10d)
Proposition.
is even a C - E map, see J. L. Taylor [Tay- i].
An infinite dimensional Whitehead Theorem fails in
proper homotopy theory.
Proof.
Let
Tel E
be the infinite mapping cylinder (telescope)
Tel (,<---- E0~--- EI~--- ...);
define
Tel B
and
Tel p:
Tel E
>Tel B
202
similarly.
Tel p
Then
isomorphism at {pn}:{E n } ~
|
is an ordinary homotopy equivalence and a
(E(Tel p):r
E)
up to homotopy where
{Bn }
not a proper homotopy equivalence
(5.5.11)
Example
[E -H -&].
simplicial sets such that not know whether
X
> ~(Tel B) ~
pro - ~,
is given by
is the ends functor) but
Tel p
is
(at = ).
We will construct below an inverse system
pro -~ (X) = 0,
but
X
is not contractible.
X
of
We do
can be chosen to be an inverse system of finite complexes or
even the end of a locally finite complex.
Let
K(Z2,n )
denote the simplicial Eilenberg -MacLane space (see [May]).
The direct system
K(Z2,1 )
Sq I > K(Z2,2)
Sq 2 > K(Z2,4)
Sq 4 ) -..
: K(Z2,2n)
~ K(Z2,2n+k)
i
has the property that all composite maps
2n+k-i = Sq
are essential (evaluate the generator of
2n+l ..- Sq
~
on the class
HI(K(Z2,1);Z2) ;
but each bonding map kills
Form the inverse system
2n Sq
x
in
H 2n
(K(Z2,1);Z2)
where
x
is
see N. E. Steenrod and D. B. A. Epstein [S -E])
*
X
2n
shown below.
203
\\\ X 3 = K(Z2,1 ) x K(Z2,2 ) x K(Z2,4) x "'"
\ \\ "\ \ \
X2 = K ( Z 2 , . ) x K ( Z 2 , 2 ) x K ( Z 2 , 4 )
kxl = K(Z2,1) • K(Z2,2) •
Then
X
Example.
m {K(w ,i)}
pro -group
~ .
contractible)
If
~ = {~ }
An Eilenberg-MacLane
such that
X
to
K(Z 2,1)
(5.5.10).
If
K(~,I)
K(Z 2,1),
= ~ n
K(Z2,2m)
using Example
X
such that
for all
n, let
X
= F n
X n --+ Xn_ I
is
Below
(5.5.11) as the
the bonding maps then there does
pro - ~I(X) = ~
and
can come from the end of a locally
question in infinite dimensional
m=l
and define twisted bonding maps
pro - spaces!
and twisting
X
{Xn }
in pro -Ho(SSo)
is not finitely dominated,
Whether such an exotic
finite complex is an~important
(i.e.,
{K(~I(Xn),I)}
by (roughly)
not exist an inverse system of finite complexes
F
X = {X n}
There exist exotic Eilenberg-MacLane
fibre of a fibration over the standard
is contractible.
we shall call
pro - space with fundamental
pro - space
is not equivalent
we shall construct an exotic
as in Example
is a p r o - g r o u p ,
the standard Eilenherg-MacLane
will be called exotic.
Let
K(Z2,~) . . . .
has the required properties.
(5.5.12) K(~,I)
x -..
topology
x K(Z2,1)__ n
so that the diagrams
(see w
for all
n,
204
Xn
K(Z2,1 )
> Xn-i
. . . .
.
Xn
id ) K ( Z . 2 , 1 )
'
pt..__
-.>
Xn-I
>
K(Z2,2
)
and
Xn
,
Xn_ I
(5.5.13)
211~ ] K(Z2,2m )
x
y
K(Z2,1 )
x
x
> K(Z2,2m+I )
(m >= 1)
co~ute.
In diagram (5.5.13);
x
in HI(K(Z2,1);Z2)
and
y
in H 2 (K(Z2,2m ;Z21
2m are the generators, and the map 2m y x X
y x x
represents the cohomology class
H 2m+l
in
(K(Z2,2m) x K(Z2,1);Z21.
As in Example (5.5.10), we obtain an inverse system of fibratlons
hE,rid Xn
-
-
->
Xn-1.
> "'"
(5.5.14)
> K(Z2,1'~..
•
> KfZ2,1"~..
> - . .
205
(5.5.15)
Theorem
[E- H - 4].
exotic Eilenberg-Maclane
The tower
diagram (5.5.14) induce isomorphisms on Further, the bohding maps of
{X }
By construction ~i'
and
(5.5.13), the bonding maps of
{pn}
{Xn }
{Fn}
pro -space
so that
{F } n
n
If
o
that class
y r H*(Xn+k,Z 2)
in
n.
By diagrams {X } n
is an
Z2 .
K(Z2,n) .
then the map
..,
o
bond
--> X n
K(Z 2, l)
H*(Xn,Z2)
Pn
This would yield homotopy commutative diagrams
bond
cohomology
for all
Hence
pro - grouv
were isomorphic to the constant tower
and for suitable
Xn ~ Fn
are null-homotopic.
Xn+ k
n
the fibrations
are compatible.
with fundamental
would be an isomorphism.
for each
constructed above is an
pro - space.
We sketch the proof for comvleteness.
Eilenberg-MacLane
X = {Xn}
id
k
"~
depending upon
> H*(Xn+k,Z 2) maps to
~onstruction in diagram (5.5.13).
K(~ 2 ,1)
n.
Thus the induced map on
factors through 0
H*(K(Z 2,1),Z2),
in H*(Xn;Z2).
Hence,
X = {X n}
\so
This contradicts the
is exotic.
D
n
(5.5.16)
Ill
i ~ I.
Example.
Let
Hence, the map
Xn =
~ S i. i=l
h:holim [Xn}
Then
~ {X n}
pro-~.{X } i n
~s stable for
is a # - isomorphism.
206
Also,
X = {X } n
that if
X
is movable.
is movable and
But
X
is not stable.
pro - ~,(X)
Dydak [Dyd- i] has shown X
is stable.
We shall define strong homotopy and homology theories on
pro -SS,.
w
is stable, then
Strong homotopy and homology theories. At
present our main application is the development of generalized Steenrod homology theories on compact metric spaces (see w
(5.6.1)
{x.}
set)
3
Definition.
The strong homotopy groups of a pro - (pointed simplicial
are given by
~i{Xj} ~ Ho(pro-SS,)(S i, {Xj}) = Ho(SS,)(S i, holim {Xj})
~i(holim {X.}). 3
These satisfy the usual properties, in particular a fibration sequence in pro -SS,
yields a long-exact-sequence of homotopy groups.
(5.6.2) {Xj},
Strong stable homotopy groups.
For any
pro - (simplicial spectrum)
define
~'S{x'}z 3 -Ho(pro-Sp)(S i, {Xj}) -= Ho(Sp)(Si, hollm {Xj})
~iS(holim {Xj}).
(5.6.3)
Proposition.
{=.s} i
pro -Sp.
forms a generalized homology theory on
207
Proof.
Clearly the functors
7. 1
s
on pro - Sp
begin by verifying the exactness axiom.
are homotopy invarlant.
Because each cofibration
isomorphic to a levelwise cofibration
{Aj} --->{X.}3
A--->X
We is
(Proposition (3.3.36)) it
suffices to show that an inverse system of coflbration sequences {Aj --> X.3 --->Xj/Aj} > ~iS{xj }
~iS{Aj } fibrant
induces three-term exact sequences ' ~i s {Xj/Aj}.
pro - (slmplicial spectrum)
{X.s} --> {Y'3}
by a fibration
(levelwise) fibre
{Fj},
{Xj/Aj }
We may functorially replace {Yj}
(Xj/Aj
{X'j.} ~
{Y'J}
> Yj),
by a
replace the map
( % ---->X'j)
in pro - Sp
with
and form compatible commutative diagrams
A. 3
> X. J
~ x./A. 2 3
1 (5.6.4)
F_
~
X'
J
-
(The left weak equivalences arises because fibration sequences in "the same" in ~. s {A.} z
>
Ho(Sp);
Ho(Sp).)
~iS(xj }
>
~i s
3
is exact on both coflbration and
Therefore the sequences
~is(xj/Aj}
and
=.S{F.) z 3
>~. s {X , .} z 3
>~.S{y.) i 3
are
But the latter sequence is isomorphic to the sequence
~iS(holim {Fj }) -->~iS(holim
] }) ---->~iS(holim {Y. J }) '
{X'.
an inverse system of fibrations is a fibration sequence in holim ~ lim
Y.
loosely, cofibration and fibratlon sequences are
j
isomorphic.
>
on the fibrant objects
(Fj},
{X'j}
preserves fibration sequences (use Theorem (3.3.4)).
and
which is exact because Ho(pro - Sp), {Y.} 3
(w
and
lim
The exactness axiom follows.
208
We may iterate this process to obtain a long-exact sequence.
For the suspension axiom, consider a cofibration sequence of the form {xj}
~ {cxj}
>{~xj}.
By regarding this sequence as a fibration sequence ,
we obtain an exact sequence
0 = ~i+l s {CXj} ---->~i+l S{ EXj}
hence the suspension axiom holds.
(5.6.5)
~ > ~'S{x'}l 3
~ ~iS{c Xj} = O;
The conclusion follows.
D
Strong homotop~ groups and homotopy pro-groups.
Both unstably and
stably these are related by the Bousfield-Kan spectral sequence (w
(5.6.6)
Strong (ordinary) homology groups.
with identity.
*
X
in SS,,
(RX) n
is the hasepoint of
RX = {(RX)n,di,si} , cofibration sequence RA--->RX--->R(X/A),
R
is the free R -module with X.
be a commutative ring S~,(;R)
on pro -SS,.
a free simpliclal R - m o d u l e Xn
functor;
as basis, mod
R*
where
There results a simpllcial R " module,
which depends functorially on A--~X--->X/A the functor
in SS, ~,(R-)
We prolong the Bousfield-Kan functor
X.
R
maps a
is a (reduced) homology theory on
R
[D - T],
to pro - SS,
and define
SH,({Xj};R)
Because
into a fibration sequence
As with Dold and Thom's infinite symmetric product
R{Xj} = {RXj},
R
We shall develop a strong homology theory
Bousfield~ and Kan [B - K] associate with for
Let
= w,(R{Xj}~.
~,(R-) ~ fi, C-;R). by defining
SS,.
209
Observe that
R
takes an inverse system of cofibration sequences into an
inverse system of fibration sequences, which is a fibration sequence in Ho(pro -SS.)
by Proposition (3.4.17).
This yields the exactness axiom for
S~,( ;R)For the suspension axiom, first observe that natural map (induced by the identity of E
R)
yields maps
{EXj }
is the simplicial suspension (see e.g. [May]).
{Xj}
> R{Xj }
9> {ERXj } ---->{WRXj }
But
{WRX.}
3
where
is a
pro -(simplicial R -module) so we obtain a map
R{EXj } - {REXj } ---->{WRX.j}.
It is easy to check that this map is a level weak equivalence.
Hence
S}{,({Xj};R) _--~.{%}
,~+!{WREj 9
(use the flbration sequence {RX.}
> {WRX.}
3
> {laX.}
3
3
(3.4.17), and (5.6.1)).
Therefore,
SH,(-;R)
is a homology theory on
pro -SS,.
On
SS,,
Sfi.(-;R) ~ h.(-;R). (5.6.7)
Strong (generalized) homology groups.
reduced homology theory on
CW,,
Let
h,
which is represented by a
be a generalized CW
spectrum
E.
210
That is,
E = {Enln ~ 0},
together with cellular inclusions
E En---->En+l,
and
h,(X) = ~,S(x A E) E ~,S{x A En}
on
CW,.
pro- Sp
We prolong the smash product
- AE
to a functor from
pro- SS,
to
by defining
{X.}3 ^ E = {Sin ((RXj) ^ E)}, and set Sh,{Xj} = ~,S({xj} ^ E).
Because
- ^E
sequences over Sh,{Xj}
takes cofibration sequences over Ho(pro -Sp),
and similarly,
Ho(pro -SS,) - AE
into cofibration
preserves suspensions,
is a generalized reduced homology theory on
pro -SS,.
As above,
S~. ~ ~, on SS,. It is now easy to see that any map of spectra
E--+F
transformation of strong generalized homology theories E and F
represent
h,
and k,
and products associated with
(5.6.8)
Proposition.
h,
For
respectively. extend to
{X.} 3
s~.
in SS,,
induces a natural
S~, ____>S~,
where
Thus all homology operations (see [Adams -i, 3], also (2.2.65~.
there is a Bousfield-Kan spectral
sequence
E2p,q = limPj{hq(Xj)},
which converges under suitable conditions to
Sh,{Xj}.
If
{Xj}
is isomorphic
211
to a tower in
Ho(pro -SS,),
the spectral sequence collapses to the short exact
sequences
0
Proof.
~ limi{hn+l(Xj )}
Use (5.6.6),
' > StUn(X)
> lira {hn(Xj)}----->
O.
(5.6.7) and the Bousfield-Kan spectral sequence (4.9.~.
We conclude this section with several remarks about cohomology theories.
(5.6.9) theory on
Cohomology groups. SS,,
then
h
If
~* h
is a generalized,
induces a cohomology theory on
reduced cohomology p r o - SS,:
~, ~~ h {Xj} -- colimj{h (Xj)},
because
colim
is exact.
Further, if
En
represents
~n
on SS,,
then
~* h {Xj} = pro-Ho(SS,)({Xj},En)
-= Ho(pro -SS,) ({%},En).
The latter isomorphism exists because
(5.6.10)
Representable
E
theories.
is stable.
n
For
{Yk } E p r o - S S ,
there is an
associated generalized cohomology theory defined by
~n(_) = Ho(pro- SS,)(-,~-n{Yk}),
Conversely, Alex Heller [Hel -i, w on an
h- e
showed that any group-valued cohomology theory
category (an abstraction of
categories as abstractions of
SS,)
n ~ 0.
CW,
analogous to pointed closed model
is the colimit of a directed system of
212
representable theories. stable category (such as
Because Heller's proof only uses factorization through a pro -Sp),
cofibrations, his result also holds for
and properties of the homotopy relation and pro -SS,
and
pro- gp.
w
w
PROPER HOMOTOPY THEORY
Introduction. In this section we shall use pro -homotopy theory to study the proper homotopy
theory of locally compact, a -compact Hausdorff spaces via a functor (the end) into pro -Top.
This functor describes proper homotopy theory at
~
We obtain
proper homotopy theory by combining proper homotopy theory at
~
with ordinary
homotopy theory.
w
contains the basic definitions of proper homotopy theory and the end
functor.
In w
we shall prove that the end functor from locally compact, o -compact
Hausdorff spaces to
pro -Top
yields a full embedding of the proper category
at
We discuss proper Whitehead Theorems in w
In particular we sketch a proof
of L. Siebenmann's finite-dlmensional proper Whitehead Theorem [Sieb -i] and show that an infinlte-dimenslonal analogue (claimed by E. M. Brown [Br] and
F. T. Farrell, L. R. Taylor, and J. B. Wagoner [F - T -W]) fails in general.
We discuss the Chapman complement theorem and an analogous "strong" complement theorem in w
214
w
Proper homotopy and ends. We shall show how T. Chapman's
[Chap -I] formulation
leads to an embedding of the proper category at
(6.2.1) f:X--+Y
Proper homotopy
A c X
This is just a reformulation f,g:X-->Y
with
theory
into a closed model category.
theory following Chapman.
o f locally compact Hausdorff
there is a compactum
~
of proper homotopy
Call a continuous map
spaces proper if for each compactum
f(cl(X \A))
c cl(Y \ B)
B c Y
(cl denotes closure).
of the usual notion of proper map.
Proper maps
are called properZy homotopic if there is a proper homotopy
H:X x I
~Y
with
Hi0 = f
and
HJl = g.
Proper-homotopy-equivalences
are now
defined in the obvious way.
Chapman also introduced weak proper homotopy
theory for the complement
theorem.
See w
(6.2.2)
Definition.
Let
spaces and proper maps, and Let
Po
and
Ho(Pc)
P
be the category of locally compact Hausdorff
Ho(P)
be its associated proper homotopy category.
be the restrictions
We shall study proper homotopy
to o -compact
theory by associating
spaces.
to each proper map its
ordinary homotopy class and its proper homotopy class at homotopy
theory at
=
by introducing
the following
the sense of P. Gabriel and M. Zisman
(6.2.3)
The proper category at
=
We describe proper
category of right fractions in
[G-Z].
~
cofinal if the closure of the complement of
We shall call an inclusion A,
cI(X\A),
j:A r
X
is compact; we shall
215
also sometimes say that
A
is cofinal in
inclusions in the proper category
P.
X.
Let
-=P\Z
will be called the proper category at a~erm
at
We shall sometimes call a morphism in
of a proper map.
~
P \ Z
It is easy to prove that means that each morphism from
X
admits a calculus of right fractions. to Y
X<
where
A
diagrams f, = f,,
is cofinal in X 9
~ A'
X
f' > Y
and
f
and
class of cofinal inclusions
Z
in P \ Z
f
-~A
~ A"
X.
~
>Y,
f" ~ Y
Two such
represent the same morphism if The
clearly contains identity maps and is closed under
of proper maps.
X 9
9A
~ >Y
and
Y<----=B
Form the solid-arrow diagram
c /
x
P).
Composition is defined as follows.
Suppose that the diagrams
represent germs at
This
can be represented by a diagram
is a proper map (morphism in
X (
on a cofinal subspace of
composition.
be the class of all cofinal
The quotient category
P
P
E
2
\ \ flC
Y
Z
~ >Z
216
Let
C = f-l(B).
cofinal in
X.
Because
f
is proper,
C
is cofinal in
A,
hence also
The required composite morphism is represented by
X<
~C
g ~ flC" Z.
It is now easy to make the connection between proper homotopy theory at
and
pro- homotopy theory.
(6.2.4) of
X
Definition.
Let
X
be a locally compact, Hausdorff space.
The end
is the inverse system
e(X) = {cI(X\A) [ A
a compactum in
X} ,
bonded by inclusion.
It is now easy t o
check that a map
spaces is proper if and only if
f
f:X--->Y
of locally compact Hausdorff
induces a map
~(f) :c(X)
> ~(Y)
which makes
the diagram
c(x)
E(f)
, E(Y)
L co-,-ute. pair
(X,r
More generally, the following result holds, where
(l,c)(X)
is the
217
(6.2.5)
Proposition.
The end construction yields the following funcrors and
commutative diagrams:
P,=
r
~ pro - Top
Ho(P )
e
~ Ho(pro-Top),
p
(l,c)
Ho(P)
(1,e)
:'(Top, p r o - T o p )
> Ho(Top, pro -Top)
The proof follows i m e d i a t e l y from the definitions.
(6.2.6) (I,E):P
Proposition.
~ (Top, pro -Top)
The functors
E:P
9 pro - Top
and
are full embeddings.
Again, this is clear from the definitions.
In the next section we shall prove the following.
(6.2.7) (l,e):Ho(Po)
Proposition.
The restrictions
- ~Ho(Top, p r o - T o p )
e:Ho(Pa,~)
are full embeddings.
> Ho(pro -Top)
and
218
(6.2.8)
Corollary.
Let
f:X-->Y
a - compact Hausdorff spaces. if
f
at
=
Then
f
be a proper map of locally compact, is a proper homotopy equivalence if and only
is an ordinary homotopy equivalence and a proper homotopy equivalence D
(6.2.9) (a)
Remarks.
The above results summarize our approach to proper homotopy theory. We use pro -homotopy to study proper homotopy theory at then blend proper homotopy theory at
=
= ,
and
and ordinary homotopy theory
to obtain proper homotopy theory.
(b)
The proof of Proposition (6.2.7) relies heavily on the telescope of a tower described in w towers.
i.e., on the coherent homotopy theory of
A suitable theory of coherent pro- homotopy should yield
an extension of Proposition (6.2.7) and its corollary to all of
(6.2.10)
Definitions.
weakly properly homotopie at in
pro -Ho(Top).
wHo(P)
We shall call germs at =
of proper maps
if the induced maps
e(f), e(~)
f,g:X
P
~Y
are equivalent
The corresponding weak pKoper homotopy category at
is then obtained by identifying maps in
P.
~ ,
which are weakly properly
homotopic at
T. Chapman introduced the following weak proper homotopy category in order to prove the second Dart of the complement theorem [Chap -i], see w
Call proper
maps
B
f,g:X---~Y
weakly properly homotoplc if for each compactum
in Y
219
there is a compactum B)
with
H!0 = f,
A
in X
HI1 = g,
and a homotopy and
concern was with contractible
x I --->Y
H:X
(depending upon
H(cI(X \A) x I) c cl(Y \B).
spaces.
Chapman's main
In this case, a straightforward
application
of Urysohn's lemma yields the following.
(6.2.11) Hausdorff
Proposition.
spaces.
pact and
X
to Y
and
be contractible,
classes
category to
wHo(P).
For
first let the diagram
~A
the germ of a proper map from
X
cl(X \A)
g:B
locally compact,
wHo(P )(X,Y).
X <
cl(X \ A) c i n t function
and Y
There is clearly a functor from Chapman's
the converse,
represent
X
Then there is a bijection between weak-proper-homotopy
of proper maps from
Proof.
Let
B .
f
>Y
to Y.
is compact there is a eompactum Because
> [0,i]
tracting homotopy for
B
g(aB) = 0
Y~ H :Y • [0,i] f:X-->Y
,:f(• ~(•
B
is compact and Hausdorff,
with
We may define a proper map
Because
=
and >Y
g(cl(X\A)) with
X
in X
is locally comwith
hence normal, = i.
HIo = id
and
there is a
Choose a conHII = * ~ Y.
by setting
• ~ X \
~ ~ ( f ( x ) , g(x)), x ,: A ,, I
L*,
It is easy to check that
f = f
in
x ~ X \A
wHo(P )(E,Y)
and that this construction
220
yields a well-defined weak-proper-homotopy A similar a r g ~ e n t wHo(P~
class of proper maps from
X
to Y.
shows that germs of proper maps which are equivalent in
yield the same weak-proper-homotopy
class of maps.
The conclusion
follows.
In the next section we shall prove that
> pro -Ho(Top)
c:wHo(Pc, ~)
is a
full embedding.
w
Proper homotopy
theory of c - c o m p a c t
spaces.
We shall relate the proper homotopy theory of c -compact homotopy
theory of towers, and thus prove Proposition
(6.3.1) c -compact,
The end of a c -compact Hausdorff
space.
Let
space.
spaces to the strong
(6.2.7).
X
be a locally compact,
Suppose that co
X=
where X
K 0 = @,
= cl(X \Kn)
each
Kn
for each
is compact, n
U K n n=0 and each
K n = int(Kn+l).
Define
and let
n
~'(x) ={x 0 ~ x l ~ x 2 .... }.
Then
~' (X)
is a cofinal subtower of the end of
cofinal subtowers of loosely regard
g(X)
are canonically
e' as a functor from
P
X,
isomorphic
a(X). in
to tow -Top
Of course, any two
tow - Top. and call
E'(X)
We may thus the end
C
of
X
when there is no chance of confusion.
regarded as a funetor from
Pc
Similarly,
to (Top, tow -Top),
(I,E') may be loosely
or as a functor from
Pc
to
221
the category
Filt
of filterd spaces, see w
We shall associate to a space telescope
Tel (e(X)),
(6.3.2)
see w
X
in P
with end
E'(X)
(as in (6.3.1)) the
and projection
Px:Tel (E'(X))
> X,
Px(X,t) = x.
< Px
X
Then
Px
Tel (e(X))
is a filtered map
(X
is filtered by
e~(~)).
We shall need a suitable notion of naturality for c'(X) and
a"(X)
be two eofinal towers in the end of
~'(X)
e"(X)
are mutually cofinal,
and
Tel (e-V(X)) z Tel (E"(X)) is true.
Maps
in
Ho(Tel)
H0 = f0'
homotopy.
If
and
pHt = f
f0'
and
H
fl"
Somewhat more are called
px f ~= Pxfl = f:W--->X H = {Ht}:W x [0,i]
for all
t.
We call
are also filtered maps,
called f iltered-ver tically-homo topic.
Let
Because
~s in Proposition (3.7.13).
fo,fl:W ----+,Tel (E'(X)) with
HI = fl'
X.
PX"
there is natural equivalence
vertically homotopic if there is a homotopy with
Tel (~'(X)) and
f0
)Tel (E(X)) H
a vertical
and f l
It is easy to prove the following.
are
222
(6.3.3) a)
Le~mm.
Tel %'(X))
and
are canonically equivalent up to
Tel (c"(X))
filtered vertical homotopy. b)~ The map
PX
is natural in
X.
For part (a) use the proof of Proposition (3.7.13).
Proof.
Part (b) follows
immediately.
(6.3.4) s:X
Definition.
> Tel ~'(X))
(6.3.5) maps
with
hn
(~Kn) = n -i
map
h:X--->R +
for
there results a map
9 [n- 2, n -i] n => 2.
c [n-2, n-l] s:X
s'
Filt.
hn( Kn_ I) = n - 2
and
denotes the set of non-negative real numbers) such for
n ~i.
) Tel ~'(X)),
Because
(Xn_l\Xn)
c [Kn\Kn_lJ ,
given by the formula
s(x) = (x, h(x)).
is a proper section for e'(X).
for E'(X)
(6.~.7)
with
Urysohn's Lemma yields
We may glue these maps together to obtain a proper R+
(6.3.6)
s
is a filtered map
ps = id X.
(notation:
h(Kn\Kn_l)
Clearly
e'(X)
Construction of proper sections [E - H - 5].
hn:Cl(Kn\ Kn_ I)
that
A proper section for
In fact, each proper section
comes from a suitable proper map
Proposition.
X
h:X--->R +
and formula (6.3.6).
is a strong deformation retract of
Tel ~'(X))
in
223
Proof.
The required retraction and inclusion are given by
Px:Tel (e'(X)) topy from
;X
and any proper section
idTel (e~(X))
to
SPx
s
for
The required homo-
~'(X).
is given by
H(x,t,t') = (x, ( i - t ' ) t + t'' h(x)),
where
s(x) = (x, h(x)).
The arrows in the figure 'below represent
H :
I j s(X)
For each
n
choose
m > n
so that
h(Xm) c [n,~).
Then
H(Tel (E~(X))m x [0,i]) c Tel (E'(X))n,
sQ that
H
(6.3.8)
is a filtered homotopy, as required.
Proof of Proposition (6.2.7).
(l,e):Ho(P a)
> Ho(T0p , pro -Top)
corresponding assertion about Let
X,Y E Pa"
is a full embedding.
e'(X)
H
is even vertical. D
We shall show that the functor
c:Ho(Pa,~)
Choose 'tends"
Note that
The verification of the
> Ho(pro -Top) and
~'(Y)
in
is easier and omitted.
tow -Top
By Proposition (6.3.7), proper homotopy classes of proper maps from
as in (6.3.1). X
to Y
are
224
in bijeetive correspondence with filtered-homotopy classes of filtered maps from Tel (e'(X))
to
Tel (E'(Y)).
By Proposition (3.7.19), which states that fil-
tered maps of telescopes yield a geometric model of
Ho(Top, tow -Top),
the
latter class of maps is in bljectlve correspondence with
Ho(Top, tow-Top)(e'(X),e'(Y))
see (6.3.1).
(6.3.9)
! Ho(Top, pro-Top)((X,
The conclusion follows.
Proposition.
E(X)),(Y, E(Y))),
D
The functor
e:wHo(P) ----*pro - Ho(Top)
is a full
embedding.
Proof. sider
e
is an embedding by construction.
e(X), ~(Y) e pro -Top.
e(X), e(Y)
as in (6.3.1).
Choose "ends"
e
e'(X), e'(Y)
Then
= tow-Ho(Top)(e'(X),e'(Y)),
Ho(pro -Top)(e(X),e(Y))
= Ho(tow -Top)(e'(X),e'(Y)).
Ho(tow -Top)
(5.2.3)), we may realize any map in map in
Ho(tow -Top).
above.
Q
Proposition.
is full, con-
cofinal in
pro-Ho(Top)(e(X),e(Y))
Because the functor
(6.3.10)
To show that
~ tow -Ho(Top) tow-Ho(Top)
is surjective on maps (see from
e'(X)
to e'(Y)
by a
The conclusion follows from Proposition (6.2.7) proved
The natural functor
a bijection on isomorphism classes of objects.
HO(Po, ~)
> wHo(Po,~)
induces
225
Proof.
Use the above propositions
pro-homotopy
(Corollary
(5.2.17)).
and the corresponding
D
The above result is close to a "Whitehead-type" theory;
w
Theorem in proper homotopy
see w
Whitehead - type TheOrems. L. Siebenmann
[Sieb] gave various convenient criteria for a proper map of
finite-dimensional,
one-ended
(see (6.4.1))
be a proper homotopy equivalence. "homotopy groups at
~ ."
three positive results:
(6.4.1)
Basepoints.
criterion implicitly
we showed that Siebenmann's
restriction
Siebenmann's;
involving weak equivalences
locally finite simplicial
Siebenmann's
In w
without a flnite-dimensional
(see (5.5.10d)).
follows.
Let
so basepoints embedding.
X for
be a non-compact
e(X,m) =
pro -Top,.
Call
X
to
(cs
Note that
one-ended
space in
are irrelevant).
We associate
to
involves
We shall discuss here and a result
at
We shall need basepoints
c(X)
complexes
criterion fails
a result involving movability;
in order to obtain the p r o -
homotopy groups of the end of locally compact space.
in
result for towers in
X
and m
P Let
(if
We introduce "basepoints" X
were compact,
m:[0,=)
~X
~(X) = ~ ,
be a proper
the inverse system
u ~[0,~),m(0))[A
E(X,m) ~ e(X)
in
a compactum in
as
X}
Ho(pro - Top).
if there is a unique proper homotopy class of proper maps
226 [0,=) --~X
(6.4.2)
in
Ho(P),
(equivalently, in
Homotopy "groups" at
=
Ho(P )).
These are the pro - groups
pro -wi(e(X,~)),
i = 1,2,''',
and
pro - ~,(e(X,~)),
where
~,(-) - H
~i (-) - [V i=l
si,-]; i=l
see w
In (5.5.10d) we constructed a proper map
p
of infinlte-dimenslonal,
one-ended
countable, locally finite simplicial complexes which was an ordinary homotopy equivalence and induced an isomorphism on
p r o - ~,,
but was not a proper homo-
topy equivalence.
However, by introducing suitable dimension restrictions or movability assumptions, one obtains the following positive results.
(6.4.5)
Theorem.
Let
f:X--->Y
be a proper map of one-ended, connected,
countable, locally finite simpllcial complexes which is an ordinary homotopy equivalence and induces isomorphisms
pro - ~,e(X)
~ pro- ~,e(Y).
Then
f
a proper homotopy equivalence if either of the following additional conditions holds. a)
[Sieb]
b)
f
dim X < =
is movable.
and
dim Y < | ;
is
227
Proof.
By Corollary
is invertible e(X)
and
in
(6.2.8),
Ho(pro -Top).
e(Y)
it suffices Because
X
admit cofinal subtowers
Theorem (5.5.6), with basepoints
and Y
E'(X)
Q -manifolds,
~(f)
are countable,
E'(Y).
and
defined as above.
We now describe a useful substitute work on compactifying
to verify that the induced map
Now use
D
for a true Whitehead
Chapman and Siebenmann
Theorem.
In their
[C - S] asked the follow-
ing questions. i)
Is every weak-proper-homotopy
equivalence
a proper homotopy
equivalence? 2)
Is every weak-proper-homotopy homotopic
equivalence weakly-properly-
to a proper homotopy equivalence?
Chapman and Siebenmann confirmed a special case of (2), namely the case of Q-manifolds
with tame ends.
We obtained
the general case in [ E - H -3].
It is
much easier to obtain the following result, which in fact suffices for the applications in [ C - S ] .
(6.4.6) invertible
Theorem. in
Let
f:X--->Y
pro -Ho(Top)
a proper homotopy equivalence and
g = f
Proof. such that
in
and
f
be a map in
P
is invertible
in
g:X--->Y
such that
o
such that
e(f)
Ho(Top).
is
Then there is
e(g) = e(f)
in
pro -Ho(Top)
Ho(Top).
By Theorem (5.2.9) g' = e(f)
Use Theorem (6.3.8)
in
there is a map
pro -Ho(Top)
to realize
g'
and
as a map
g':E(X) g'
> E(Y)
is invertible
g" : (X \ K0)
~Y
in in
p r o - Top Ho(pro-Top).
for some
228
compact,-,
K 0 cX:
Ho(Top).
Let
--->[0,I]
E(g") = g'.
H:(X \Ko) x [0,i]
choose a compactum h: 5
i.e.,
with
K1 c X
with
9Y
For suitable
be a homotopy from
K 0 cint K 1
h(K O) = 0
and
r
E(x),
K0,
g" = fI(X\K0) f
to g".
in
Then
and Urysohn function
h(bd KI) = I.
The required map
g
is given
by
for
x e K0,
g(x) = ~H(x,h(•
for
xr
],, k g C:4)9
for
x ( X \ K I.
The required properties are easily verified.
KI,Ko,
D
The following chart smmnarizes our use of pro- homotopy theory.
Proper homotopy
Pro- homotopy
theory at
Strong
Weak
theory
Ho(P )
Ho(pro-Top)
wHo(P|
pro- Ho(Top)
We shall see further connections in the next section.
w
The Chapman Complement Theorem. In the late 1960's Borsuk sparked an avalanche of interest in the study of the
global homotopy properties of compacta (see [Mar -3], surveys of shape theory).
[Ed -i] and [E -H -2] for
Borsuk's original formulation of the shape theory of
229
compact subsets of Hilbert space [Bor- 2] lacks the flexibility of the approach to be described in w
but it has the advantage of being more geometric.
This
added geometry was quickly capitalized upon by Chapman in [Chap -i].
Let
s = H
(-l/n, l/n)
be the psuedo-interior of the Hilbert cube
n=l Q = ~
[-l/n, l/n].
One defines the fundamental category or shape category,
n=l Sh,
as follows.
The objects of
are compact subsets of
a sequence of maps
V
of Y
in Q
n,
V.
Note that
near
Y.
and
then a fundamental Sequence
fn:Q--->Q
n' ~ n o
fn(X)
U
fn, IU
of X
in Q
fnlU
and
the restrictions
f,
f':X ~ Y
s
and Y
is defined as
and an integer
are homotopic in
V.
V
no
no
are homotople in
Y ; it only has to be
are considered homotoplc,
of Y
in Q
such that for
The morphisms in
homotopy equivalence classes of fundamental sequences.
contained in
X
and an integer
f~IU
does not have to be contained in
in Q
If
f:X--+Y
of X
provided that for every neighborhood
U
s.
with the property that for every neighborhood
Two fundamental sequences
neighborhood
fnIU
are compact subsets of
there exists a neighborhood
such that for
f = f',
s,
Sh
Sh
there exists a
n ~ no,
are now taken to be
Two compacta
X
and Y
are said to have the same shape if they are isomorphic in
In [Chap -i] Chapman proved the following beautiful theorem.
Sh.
230
(6.5.1) X
and Y
Chapman Complement TheqKem.
If
X
and Y
are compacta in
have the same shape if and only if their complements
Q \X
s,
and
then
Q \Y
are homeomorphic.
Chapman then extended the association
X ~ - > Q iX
to a functor
shape category to the weak proper homotopy category of complements in pacta in
T
from the Q
of com-
s.
(6.5.2)
Definition.
Let
PQ,~
Q
of
compacta in
s
and germs at
Because
Q
is contractible, Chapman's weak proper homotopy category above is
isomorphic to
wHO(PQ ~)
|
be the category of complements in
of proper maps.
(use Proposition (6.2.11)).
Chapman then proved the following categorical version of the Complement Theorem, stated in our language.
(6.5.3)
Theorem.
Outline of proof. U = {U.} J tively, in
and Q.
V = {Vk}
There is a category isomorphism
Let
K
and L
be compacta in
T:Sh
s.
be bases of open neighborhoods for
Chapman showed that
K
and L
inclusions
{Uj \ K} ~ ~{Uj} and
{Vk\ L} ~ { V k}
> wHO(PQ,~).
Let K
and L,
respee-
are Z -sets, hence the natural
231
are levelwise homotopy equivalences. Sh(X,Y) a pro -Ho(Top)(U,V).
It is easy to show that
We then obtain a string of isomorphisms
Sh(X,Y) ~ pro-Ho(Top)(U,V)
a pro -Ho(Top)({Uj \ K},{V k \ e})
wHO(PQ,~)(Q\ K, (the complements of
(6.5.4)
V. 3
and U k
are compact in
Th___~estrong shape category.
easy to check that the complement countable basis of open sets
Uj
Q \K with
Let
s
in
Q;
K
Q).
The conclusion follows.
be a compactum in
s.
is ~ - compact, and hence that cI(Uj)
therefore define the strong shape category in
Q \ L)
c5_
s - Sh
1
K
(j = 0,i,2,-'').
has a We may
to be the category of compacta
and coherent homotopy classes of maps of their associated neighborhood bases namely
(6.5.4)
s - Sh(K,L) E Ho(Tel)(Con Tel {U.}, Con Tel {Vk}) 3 m Ho(Tel)(Con Tel {cl(Uj)}, Con Tel {cl(Vk)})
where
It is
Con Tel
see w
(6.5.5)
is the contractible telescope
By Proposition (3.7.20),
s-Sh(K,L)
~ Ho(pro-Top)({Uj},{Vk}).
232 By following the
proof of Theorem (6.5.3), we obtain the following strong
categorical version of the Complement Theorem.
(6.5.6)
Theorem.
There is a commutative diagram of categories and functors
s-
Sh(K,L)
Sh(K,L)
(6.5.7)
Remarks.
9 HO(PQ ~)
i
> wHO(PQ|
E
Wlth reference to the orlginal Chapman Complement Theorem
(6.5.1), recall that the functor
HO(PQ~)
> wHO(PQ|
isomorphism classes of objects (Proposition (6.3.10)).
induces a bljection on
w
w
GROUP ACTIONS ON INFINITE DIMENSIONAL MANIFOLDS.
Introduction. We shall discuss the classification
s =
H (-i/n, I/n) n=l
In w
and on
Q =
of actions of compact Lie groups on
~ [-l/n, l/n]. n=l
we shall review the theory of s - m a n i f o l d s
Standard group actions will be constructed
and Q -manifolds.
in w
We shall also show that all principal actions on
s
following Jim West
[West -i].
are standard up to equivariant
homeomorphism.
w
contains a classification
actions of a finite group on interesting,
w
theorem
Q.
Whether such actions are unique is an open,
and deep question.
Basic theory of s - m a n i f o l d s An s -manifold
homeomorphic
(7.2.1) Anderson's
(largely due to West) for semifree
(respectively,
to an open set in
s -manifolds.
and Q - manifolds.
Q - manifold) s
is a separable metric space locally
(respectively,
Q).
Work of Kadec, Bessaga,
and Pelczynski
proof that all separable Frechet spaces are homeomorphlc
The classification
theory of s -manifolds
is due to Henderson
culminated
in
(see [A -B]).
[Hend].
First,
This section represents joint work with Jim West and much of the material taken from his unpublished notes [West-i].
is
234
every s- manifold simplicial
M
complex
can be trlan$ulated, K
such that
homotopy equivalence
M
f:M 1 --->M2
that is, there is a locally finite
is homeomorphic
to
K x s.
between s- manifolds
Second, every
is homotopic
to a homeo-
morphism.
Torunczyk X
is an
[Tow -1,2] has shown that
ANR
(7.2.2)
~ -manifolds.
[Kel] showed that
[West -2,3].
Q
West
Q
complexes
K.
Chapman
K x Q
If a map
to a homeomorphism
f:K--->L
thus every compact
ANR
to
K x Q
ANR's;
f x IdQ:K • Q
type of a locally finite simplicial
complex.
is
is
[Chap- 3]
torsion.
ANR
has finite homotopy
every such
~L x Q
The converse also holds
of Whitehead
in particular,
K
for some locally
is the CE -image of a type.
Chapman
that West's result can be used to extend = - simple homotopy
locally compact
if
of locally finite simplicial
then
[West - 2].
invariance
[Chap -1 - 7] and
is a Q - m a n i f o l d
i.e., homeomorphic
[West - 3] showed that every locally compact
Q -manifold;
(see w
[Chap - 4] proved the converse:
is an ~ -simple homotopy equivalence,
and implies the topological
observed
complex.
manifold is trian~ulable,
properly homotoplc
West
is due to Chapman
[West -2] showed that
finite simplicial complex
if
is homogeneous.
theory of Q - manifolds
a locally finite simplicial every
if and only if
Many of the above results hold for Q - manifolds
is replaced by ~ -simple homotopy equivalence
The classification West
is an s - manifold
(separable, metric).
homotopy equivalence Keller
X x s
ANR
[Chap - 7]
theory to
has the ~ -simple homotopy
Hence, a proper map
f:MI--->M 2
235
between Q - m a n i f o l d s
is properly homotopic
to a homeomorphism
if and only if
f
is
-simple.
Bob Edwards only if)
X
[Edw] has recently shown that
is a locally compact metric
X • Q
(7.2.3)
[West -i]
treatment of
theory after Siebenmann. ''~
The basic idea of (finite)
simple homotopy
theory
[Cohen],
single-out and study m a p s of finite cell complexes which are homotopic compositions
of maps which are of the form of an inclusion
.n-i on K = L u e u e
with
e
n-i
an n - c e l l
locally finite and n o t necessarily i
above by an inclusion
of disjoint complexes (K~ n L)
> Ki
Ki
S(K)
each of which collapses
S(K).
In his treatment the group
is an equivalence
with domain
K,
complexes, K \L
to
Kin
where
this notion to
one replaces
is the union L,
the u
K. i
i.e., such as
i,
and
to proper mappings and proper homotopies.
Siebenmann introduces of
in which
to finite
or are of the
is the result of a finite sequence of inclusions
one restricts oneself
(7.2.4)
j:L --~K
n e ,
In generalizing
finite-dimensional
is to
i:L--->K,
which is a face of
form of a homotopy inverse to such an inclusion.
map
if (and
ANR.
We shall conclude this section by giving West's "infinite simple homotopy
is a Q - m a n i f o l d
where
class
g:K--->M
S(K)
[Sieh] of infinite simple homotopy of simple structures on
[f:K-->L]
Each element
of proper homotopy equivalences
is in the class of
This survey is a quote from [West -i].
K.
theory,
f
whenever
there is a
236
simple homotopy equivalence tion on
S(K)
s:L ---~M
such that
[f] and [g]
tops (domains) of mapping cylinders, and then
K.)
(The group opera-
need not concern us here, but it is (essentially) geometrically
defined in [Sieh] by representing
inclusion of
g = sf.
K
with inclusions, say, into the [f] [g]
is represented by the
into the result of identifying the two spaces along the copies of
In particular, if there is only one simple structure on
K,
then all proper
homotopy equivalences are simple.
(7.2.5)
Wh,
head functor
Wh
K0,
and limits.
K.
is that these are functors.) K
S(K),
Siebenmann uses the White-
and the projective class group functor
constructions at the end of
end of
To examine
K0
in several limiting
(In essence, all that is needed for this paper The limiting constructiens are as follows.
Let the
be
e(K) = {K = W 0 ~ W I m W 2 ~ --'},
(see (6.2.14)), where the
W
are subcomplexes of
K
whose complements have
n
pact closure.
Choose a proper base ray
e(K,~) = {(Wn u m[O,=),m(O))}.
a:[0, =) -
~K
as in w
Now consider the inverse system
Tie(K) m ~iS(Kl,m) ~ {nl(Wn, m(0))}.
Now define
K 0 ~IE(K) E limn{K 0 ~l(Wn, ~(0)), K0(i,)},
Wh ~le(K) 5 limn{Wh ~l(Wn, ~(0)), Wh(i,)},
Let
com-
237
and the attenuation
Wh ~le'(K) E limln{Wh ~l(Wn, m(0)), Wh(i,)}.
Observe that the attenuation is zero if
~le(K)
is stable, i.e., p r o -
isomorphic to a group.
Exact sequences. S(k).
Siebenmann gives two exact sequences to aid in computing
They are as follows:
(7.2.6)
0
(7.2.7)
(Sb(K)
~Sb(K)
Wh TIe(K )
> S(K)
~ W h ~I(K)
> K 0 ~le(K)
~Sb(K )
~ K 0 ~I(K);
~ W h ~I~'(K)
> 0.
is the group of equivalence classes of proper homotopy equivalences defined
analogously to
S(K)
but where one allows the inclusion
K~ n L
>K.
to be any
inclusion of finite complexes which is a homotopy equivalence.)
w
The Standard Actions. Following West [West -i], we shall construct standard principal actions of any
compact Lie group
G
on Q0
that all principal actions of
left translation.
(Q
G
~ C(G) i=l
on s
are standard.
Let
G
s.
We shall show
act on itself by
This principal action extends to a semi-principal action with
unique fixed point on the cone of
action on
with a point deleted) and on
G,
C(G) E G x [0,i] / G x {0}.
The product
is also semi-principal with unique fixed point, the infinite
238
cone point.
Q.
aG
H C(G) iffil
is homeomorphic to
Removing the unique fixed point yields the standard principal action
of G
Q0 • s
PG
But, it follows from [West -2] that
on Q0"
Since
Q0
is homeomorphlc to
of G
on s.
is contractible, and
s.
Q0 x s
is an s -manifold,
Thus, we also obtain a standard action
Any principal action of
G
on Q0
or s
which is not con-
jugate to the standard action will be called exotic.
Let
p
and p'
be two principal actions of
the actions are nice if the quotient spaces (e.g., if s/~
G
are both classifying spaces for
map such that the induced bundle
s/p
on s.
(s,p)
and s/p'
(see [Hus]).
f (s,p') Since
f
are both S -manifolds,
G.
over
We will say that
and s/p'
is finite, then this is always the case).
and s/p'
bundle
s/p
G
are s -manifol~s
In any ease, Let
s/p
f:s/p
~s/p'
be a
is isomorphic to the
is a homotopy equivalence, if
then
f
is homotopic to a homeomorphlsm
g 9
One thus obtains the diagram of principal G -bundle isomorphisms
S
)
S
~
S
(7.3.1)
slp
in which hence
h=~or p
and O'
> s/p
$
"-slp'
is a G -equivariant homeomorphism from are equivalent actions of
G
on s.
(s,p)
to (s, p');
Summarizing, we have the
239
following theorems.
(7.3.2)
Theorem.
are standard.
(7.3.3)
The
G
on s
0
Theorem.
standard.
All nice principal actions of a compact Lie group
All free actions of a finite group
G
on s
are
0
Qo
case is much more subtle~ since one must show that
q0/p
and
Q0/p '
have the same = - simple homotopy type, and not just the same homotopy type, before one can conclude that they are homeomorphic.
The main result of this section is
the following theorem, which will be proved in w
(7.3.4) G
Theorem.
on Q0"
Let
p and p'
be free actions of a finite group
Then the following statements are equivalent:
i)
p
is equivalent to
2)
Q0/p
is homeomorphic
3)
Qo/p
is ~ -simple homotopy equivalent to
4)
Q0/p
is proper homotopy equivalent to
5)
The end of c(Q0/p')
(7.3.5)
in
Remarks.
to West [West -i]. c(Q0/p)
Qo/p,
p'; to
Q0/p';
E(Q0/p)
Qo/p'; Q0/p';
is homotopy equivalent to
pro -Ho(Top).
The equivalence of (1) - (4) for The end
~(Q0/p)
G
is a quotient of
is a pro -space analog of the classifying space
showed that the natural inclusion
~(Q0/p)
~ Q0/p
a finite group is due ~(Q0 ) = pt.;
hence,
BG = Q0/p.
West
is always a~ - isomorphism;
240
even an isomorphism on if and only if exotic
p
is standard.
K(Z2,1)'s;
actions on
Q0"
pro -~,.
By Theorem 7.3.4, it is a homotopy equivalence In w
but we still do not know of any exotic compact Lie group On the other hand, work of Tucker [Tuc -2] shows that there are
uncountably many different actions of
w
we showed the existence of uncountably many
Z
on QO"
Proof of Theorem (7.3.4). The following preliminary lend,as, as well as the equivalence of statements (i)-
(4) in (7.3.4) are taken from [West -i].
Our machinery (pro - spaces) is used to
simplify some of the statements and arguments.
Let point
G
be a fixed finite group acting semifreely on
q.
Q
with unique fixed
Let
a:C • Q
(g,x) ~ a~ gx
9 Q,
denote the action.
(7.4.1) tractible (in
Proof.
Lepta.
Q \ {q}
is contractible, and its end
g(Q \ {q})
is con-
Ho(pro -Top)).
Represent
Q
as the product
[0,i] i.
Because
Q
is homo-
i=l geneous (see 7.2.2), we may assume that vex, hence contractible.
q = (0,0,0, ....).
Q\{q}
is then con-
Also,
~(q\{q}) ~ {ui ~
(O,l/i]i ~
n
(o,1]jliil},
j >i
bonded by inclusion.
Because each
U. l
is convex, hence contractible,
r
\{q})
241
is contractible in (5.2.17).
pro -H0(Top),
hence in
Ho(pro -Top)
0
(7.4.2)
Lemma.
There is a commutative diagram of covering maps in
G
. . . .
(7.4.3)
I
e(q \ {q}) - -
~ Q \{q}
1
1
e((Q \{q})/~)
Proof.
' (Q\ {q})/u.
First, choose a representative
U 0 = Q\ {q}
and
pro - Top
G
I
Let
by Corollary
{Ui[i ~ i}
tower for
E(Q \{q})
be as in (7.4.1).
For each
as follows.
n gU i
i,
gcG (the intersection of translates of
Ui
under
u)
contracts
in
u. \ {q}.
Let
l
(7.4.4)
~ ( Q l { q } ) _-- {vo = v I = v 2 . . . .
be a subsequenee of
that each
V.
{
0 gEG
gUi}
chosen so that
Vi
}
contracts in
is invariant under
1
Next, rewrite diagram (7.4.3) as
-
(7.4.5)
l
c(Q \ {q})
l
-
G
i
Q \ {q}
~(p)
e((Q \ {q})/e)
> (Q \ { q } ) / a,
Vi_ I.
Note
242
where
p
,
is the covering map induced by
and
c(p)
is the levelwise covering
map
E(Q\ {q}) = {V.} l
yielding the conclusion.
(7.4.6)
D
Corollary.
pro -Wl ffiG;
{V.la} ~ e((Q\ {q})l=), I
E((Q\ {q})/=)
is an Eilenberg-MacLane pro - s p a c e with {(Vi/~ ) }
i.e., the tower of universal covers
is contractible in
Ho(pro -Top).
Proof.
It remains only to observe that
easy exercise involving covering spaces.
(7.4.8)
Proof of Theorem (7.3.4).
(i) ~ ( 2 )
{(Vi/~ ) } ~- ~ ( Q \ { q } ) ;
this is an
0
The following implications are easy:
by covering space theory;
(2) ~=~ (3) by Chapman and West's classification of Q - manifolds (see (7.2.2)); (3) ~
(4) by definition; and
(4) = ~ (5) by definition.
To verify (4) --~ (3), we shall show that for any semi-free action finite group
G
on Q
with unique fixed point
any proper homotopy equivalence with domain late
(Q\{q})/=
as
K • Q
Because the projection map
(7.4.9)
q,
S((Q \ {q})/a) = 0,
(Q \ {q})/a
is | - simple.
for some locally finite simplicial complex (Q \ {q})/a
>K
of a so that TrianguK.
is a proper homotopy equivalence,
~le(K) ~ ~i K ~ G,
243
via the inclusion. (7.2;6).
Henc~
Also,
K 0 ~IE(K) = K 0 =I(K),
Wh TIC(K) a Wh ~I(K)
(7.2.5)), so that
Sb(K) = 0
To verify (5) ~
and
by (7.2.7).
so that
S(K) a Sb(K)
Wh ~la'(K) = 0 Hence
by
(by (7.4.9), see
S(K) = 0,
as required.
(4), first consider the diagram
holim e(K)
hollm K
1
(~)
induced by (7.4.5) (the homotopy inverse limit, holim, is developed in w167 and w
where
unless
i = i,
K
is as above.
in which case
spectral sequence to
hollm E(K)
By Corollary (7.4.6), pro -~l(C(K))
= G.
t0 holim e(K)
= 0
Applying the Bousfleld-Kan
yields
holim e(K) = ~ G ,
(note that
pro-~i(E(K))
is pointed and connected).
i = i,
i~l
Hence the map
holim e(K) --->K
is a homotopy equivalence by the ordinary Whitehead theorem
(K
by Lem~as (7.4.1) and (7.4.2)).
e(K)
is a in
K(G,I)
Ho(pro - Top)
by the diagram
Therefore
K
is a retract of
244
holim s(K)
) holim K \
".~
"
e(K)
p
Now, for two semi-free actions points
q
and q',
assume
\
_% "N \
> K
and p'
(Q \ {q})/p
1
\
of G
and
on Q
with unique fixed
(Q \ {q'})/p'
equivalent at
Then
E((Q \{q})/p) ~ E((Q\{q'})/p')
by Theorem (6.3.4).
This equivalence extends to the diagram
IK ~ '
E((Q \ {q})10) r
(7.4.12)
Remarks.
~
The implications
must verify that the maps covered by equivariant maps
(Q\ {q'})/p'
'> ( Q \ [ q ' } ) / o '
(Q \ {q))/0
(globally), as required.
for arbitrary compact Lie groups
and
> (Q \ {q})Ip
Theorem (6.3.3) now implies that
are proper homotopy equivalent
Ho(pro-Top)
ll l l
e((o. ,, { q ' ) ) / ~ ' )
by (7.4.10).
in
-,%
l
(7.4.11)
are proper homotopy
G
(2) <==~ (3) ~=~ (4) ~=~ (5)
with a similar proof.
(Q\{q})/p Q \ {q}
(I) ~
"- (Q\{q'})/p' > Q \ {q' }.
D
For
(2) ~
(i)
of statement (2) are
hold we
w
w
STEENROD HOMOTOPY THEORY
Introduction. This chapter is concerned with shape theory, shape functors, and Steenrod
homology theories.
In w
we discuss the Steenrod homology theory [St -i], the Kaminker-Schochet
[K- S] axioms for generalized Steenrod homology theories, and the Vietoris construction [Por -i]. homotopy theory. Sh,
and h,,
plexes.
We then use the Vietoris construction to discuss Steenrod In particular, we define canonical Steenrod and
of any generalized homology theory,
Proofs of the properties of
Sh,
h,,
6eeh
extensions,
defined on finite
CW
com-
occupy the next four sections.
D. S. Kahn, J. Kaminker, and C. Sehochet have independently obtained Steenrod extensions by different methods.
In w
we verify useful properties of the Vietoris functor.
In w
we prove that
metric spaces,
Sh,
that is, that
is a homology theory on the category Sh,
CM
is a homotopy invariant functor which satisfies
the first two Kaminker-Schochet axioms for a Steenrod homology theory. show that products and operations associated with
We give two spectral sequences converging to spectral sequence to prove that axiom.
of compact
Sh,
h,
Sh,
extend to
in w
We also
Sh, .
and use the first
satisfies the remaining Kaminker-Schoehet
246
In w
we prove functional and Steenrod duality theorems for
duality states that for a compactum
X
in Sn,
Sh,.
Steenrod
Sh (X) = hn-p-l(s n \X). P
Func-
tlonal duality is due to D. S. Kahn, J. Kaminker, and C. Schochet [K -K -S].
w
Steenrod homology theories. Let
h,
complexes.
be a generalized homology theory defined on the category of finite In this section we shall define a canonical Steenrod extension
(see (8.2.3)) of w167
-8.5.
h,
to the category
CM
of compact metric spaces.
CW
Sh,
Proofs occupy
See (8.2.1) and (8.2.2) for ordinary Steenrod homology and Steenrod
K - homology.
The problem of constructing generalized Steenrod homology theories with "good properties" was posed by M. F. Atiyah and others at the Operator Theory and Topology Conference held at the University of Georgia in April, 1975.
By "good properties"
Atiyah meant that products and homology operations extended from
h,
to Sh,.
Our extension enjoys these properties.
(8.2.1)
Duality and homotopy theories.
cohomology of a compactum
X
in Sn
Alexander duality states that the Cech
is canonically isomorphic to the ordinary
homology with compact supports of the complement defined a homology theory the ordinary eohomology of homology.
SH,
such that
Sn \ X.
SH,
SH,(X)
Sn \ X.
is canonically isomorphic to
is now called (ordinary)
Milnor [Mil -i] (see also [Sky]) showed that
Eilenberg-Steenrod
[E -S] axioms on
CM,
In [St - i] Steenrod
and further that
SH, SH,
Steenrod
satisfies all of the is characterized by
247
these axioms together with two additional axioms; namely, invariance under relative homeomorphism (generalized excision) and the strong wedge axiom (see (8.2.3)).
(8.2.2)
Steenrod K-homology.
In a series of papers [Brow], [ B - D - F -
L. Brown, R. Douglas, and P. Fillmore defined a functor for
Ext
on CM
i- 2],
by taking
, unitary equivalence classes of C - algebra extensions of the compact
Ext (X)
operators by the C - algebra of complex-valued functions on Schochet [K-S]
X.
Kaminker and
then set
SEn(X) ~ I E x t
(X),
LExt (EX),
for
n
odd,
for
n
even,
( E is the unreduced suspension) and showed that
SE,
for a generalized reduced Steenrod homology theory.
satisfies the axioms (8.2.3) L. Brown, Douglas, and
Fillmore have shown that on the category of finite complexes,
SE,
is reduced
K- homology.
(8.2.3)
The Kaminker-Schochet
[~T S] axioms.
homology theory consists of a sequence invariant functors from the category AG
A seneralized (reduced) Steenrod
h, = {hnln c Z} CM
of covariant, homotopy
of compact metric spaces to the category
of abelian groups which satisfy the following axioms.
(E)
Exactness:
if
(X,A)
is a compact metric pair, then the
natural sequence
hn (A) is exact for all
n.
> hn(X)
> hn (X/A)
248
(S)
Suspension:
there is a sequence of natural equivalences
On:hn
called suspension, where
(W)
Strong wedge:
if
E
> hn+ 1 ~ E
is unreduced suspension.
X = v Xj E lim N { v N Xj} n=l n=l
is the strong wedge of a sequence of pointed compact metric spaces, then the natural projections
X~X.
induce an J
isomorphism
h,(X) -= ~jh,(Xj).
(8.2.4)
Remarks.
but not pointed. k,
by the formula
The homology theory
A reduced theory
k,
kn(X) = ker (kn(X)
h,
is reduced (see axiom (E) above)
may be obtained from any unreduced theory
kn(*)).
We break the problem of construction Steenrod extensions into two parts.
The
first part involves approximating a compactum by an inverse system of simplicial complexes, an idea which goes back to Alexandroff
[Alex].
The second part involves
prolonging a generalized homology theory defined on the category of finite plexes to a generalized homology theory defined on ble analogues of the Kaminker-Sehochet axioms.
pro -SS
CW
com-
and satisfying suita-
This was done in w
We shall carry out the first part of this program using a Vietoris functor (see (8.2.7); this construction was first introduced by T. Porter [Por-l]) Steenrod's original construction for motivation.
after giving
The Vietoris functor also yields
249
a Steenrod homotopy type for any compact metric space.
We can construct Steenrod theories in a wide variety of other settings, example, algebraic geometry and algebraic K -theory,
(~tale homotopy
for
theory), proper homotopy theory at
by applying suitable functors into
pro - SS
~ ,
and then
following the second part of our program.
(8.2.5)
Regular cycles followin~ Steenrod
metric space,
K
and let
f:VK--~X
VK
be the set of vertices of
such that for each
many simpliees have diameter less than coefficients complex
K
Let
X
be a compact
an abstract countable locally finite simplicial complex
simplicial complex), is a map
[St- I].
in an abelian group and a regular map
f
G,
s > 0,
e .
A regular map
the images of all but finitely
A regular R - chain on
(K,f,Cq),
as above,
K.
consists of a
elf
c
C~(X;G)
based upon regular chains, and reduced Steenrod homology
on K
with coefficients
S~q(X;G)
(8.2.6) space
Remarks.
X.
Let
Lebesque number of x
U
itself)
be an open cover of
(depending upon X
f
X.
Also,
(K,f,c) q
Then there is an x
in X,
U c U .
of the vertices of
A) e U .
simplicial
One then obtains a chain complex
such that for any point
the image under
to
G.
Consider a regular q- chain
U)
with
E Hq+ I[C~(X;G)].
is contained within a single open set of K,
set
U
in
X
together with a (possibly infinite)
q -chain
q
(clf
Hence, K
X.
~ > 0
(the
the c -neighborhood
of
for almost all simplices
is contained within some open
the fact that
conceals the "local pathology" of
on a compact metric
f
maps
VK
(and no___~t K
250
(8.2.7)
The Vletoris construction [Por -i].
topological space
X.
The Vietoris nerve of
Let U ,
U E U.
(n + i) - tuple
U ~ V).
The identity map of
and V X
is the (x0,xl,-.',x n)
and
s i ( x 0 , x l , ' " , x n) = (x0,xl,"',Xi_l,Xl,Xl,Xi+l,'",Xn),
U
VN(U),
Faces and degeneracies are given by
di(x0,xl,''',Xn) = (x0,xl,'",Xi_l,Xi+l,"',Xn),
Now consider open covers
be an open covering of a
denoted
slmplieial s e t in which an n - simplex is an ordered of points contained in an open set
U
of X
for
where
V
0 ~ i ~ n.
refines
induces a canonical inclusion
U
(notation:
VN(V) e--->VN(U).
We may therefore associate to a topological space its Vietorls complex.
(8.2.2)
VX = {VN(U) IU
cal inclusions
(8.2.9)
VN(V) c
>
Proposition.
an open cover of VN(U)
when
U
<
=
X} .
VX
is bonded by the canoni-
V .
The Vietorls construction extends to a functor
V:Top----->pro - S S .
Proof.
We define
and open cover
U
f-l(u) = {f-I(uIIU VN(f-I(u)) whenever
of Y c
> VN(U) U ~ V
U)}
V
on morphisms as follows:
to a continuous map
f:X--->y
we associate the open cover of X.
Then
((x0,Xl,-'-,Xn) ,
(and hence
f
induces an obvious map ~ (f(Xo),f(xl),-..,f(Xn))
f-l(u) ~ f-l(v)),
the diagrams
such that
251
vN(f-l(v))
> VN(V)
f
I
9
i
VN(f-I(u))
commute. Clearly
>VN(U)
These diagrams induce the required morphism V
preserves
identity maps and
Theorem
[Dow].
Let
U
theorem of Dowker.
be an open covering of a topological
Then the realization of the Vietoris nerve of
X.
homotopy equivalent
We need Vietoris
in p r o - SS.
V(fg) = Vf o Vg.
We shall frequently use the following
(8.2.10)
>VY
Vf:VX
U,
RVNU,
to the realization of the Cech nerve of
space
is canonically
U,
RCNU.
nerves rather than Cech nerves in order to obtain a funetor
from compact metric spaces to
pro - spaces.
An interesting
problem is the con-
struction of a nerve that is "small" like the Cech nerve and "rigid" like the Vietoris nerve.
(8.2.11) generalized plexes. i.e.,
V
Steenrod and Cech extensions of hom01osy theories.
Let
(reduced) homology theory defined on the category of finite
By G. W. Whitehead h,(-) ~ ~,s((_)
and Cech extensions of following formulas.
[Wh] there is a
^ E). h,
See also w
to the category
[We write
CW
Sh,
(5.6.7) and the Steenrod extension;
spectrum
and (5.6.7). CM
E
h, CW
be a com-
which represents Define the Steenrod
of compact metric spades by the
for both the extension of
h,
to pro - S S
the usage should be clear from the context.)
he,
252
(8.2.12)
Steenrod extension:
Sh.(-) ~ Sh, o V
= ~,S(hollm (Sin (RV(-) ^ E)) = Ho(pro -Sp)(S , Sin (RV(-) ^E)).
(8.2.13)
~ech exs
~, (-) = i~ {h,V(-)j} = li.j{=,S(sln (Rv(-)~^ z)} = p r o ~Ho(Sp)(S
Here,
V(-) = {V(-)j}
, Sin (RV(-) ^ E)).
denotes the Vietoris functor and
simplicial spectra.
For ordinary homology these formulas become
(8.2.14)
Sfi,(-;R) ~ Sfi,(V(-);R) = w,(holimRV(-)) = Ho(pro-SS,)(S
, RV(-));
w
(8.2.15)
.,(-;R) = limj {fi,(V(-)j;a)} = lim.j {~,(RV(-)j)} = p r o - H o ( S S , XS , R V ( - ) ) .
Sp
the category of
253
In (8.2.14) and (8.2.15),
R
denotes any commutative ring with identity as well as
the free R - module functor of Bousfield and Kan [B -K -i], not the geometric reallzation functor.
(8.2.16)
Remarks.
Ken Brown [Brown] defined generalized Sheaf cohomology
theories with a similar use of slmplicial spectra and smash products.
(8.2.17)
Proof.
Theorem.
h,
is the ~ech extension of
h,.
This follows from Dowker's Theorem (see (8.2.10)).
Alternatively,
follow the proofs of Theorems (8.2.18) and (8.2.21).
(8.2.18)
Theorem.
Sh,
(8.2.19)
Theorem.
Sh, a h,
(8.2.20)
Theorem.
Products and operations associated with
(8.2.21)
Theorem.
Sh,
is a homology theory on the category
CM
of compact
metric spaces.
on finite
CW
complexes.
h,
extend to
Sh, .
is a Steenrod homology theory on the category of com-
pact metric spaces.
(8.2.22)
Theorem
(see also [ K - K -S]).
Sh,
satisfies functional and Steen-
rod duality on compact metric spaces.
We begin the proofs by verifying properties of the Vietoris funetor in w We shall prove Theorem (8.2.18) in w
using strong homology theories on
pro- SS,
254
(5.6.7).
Theorems
a Bousfield-Kan
spectral sequence for
Theorem (8.2.21)
w
(8.2.19) and (8.2.20)
there.
The Vietoris
Proposition
topic maps of spaces
i0
X x [0,1].
(announced
f,g:X ~ Y
H:X x [0,i]
and i I
K
systems, hence
RVi 0
K 0 = RVi 0
and
K I = RVi I
> Y
X
and limits, at
in
we obtain a diagram
[Por - I]).
Homo-
p r o - SS,
induces a functor
Hi 0 = f
and
Hi I = g,
as the ends of the cylinder
Vi 0 = Vi I
in Ho(pro -SS).
in p r o - Top,
We shall
i.e., a map
>RV(X x [0,i])
(the realization
Because adjunction morphisms
V
be a homotopy with
to RVi I
wise). SS,
suspensions,
of compact metric
> Ho(pro -SS).
to show that
from
CM
induce homotopic maps in
K:RVX x [0,i]
with
in w
independently by T. Porter
are the inclusions of
It suffices
define a homotopy
[Brown]) and prove
Ho(pro -SS).
V:Ho(Top)
where
(8.2.22)
We shall develop
functor on the category
in
of their Vietoris
Let
(compare
cofibration sequences,
least up to canonical equivalence
Proof.
in w
see w
functor
spaces preserves homotopies,
Vf,Vg:VX ~ V Y ,
Sh,
We shall prove Theorem
We shall prove that the Vietoris
(8.3.1)
follow easily,
id--->Sin R
functor
R
is applied level-
are natural weak equivalences
255 (8.3.2)
vx
[0,I] - > S l n
x
(RVX x [0,I])
9 Sin RV(X x [0,i])
~--
v(x x [0,i]).
Because the "wrong-way" arrow in diagram (8.3.2) is invertible in we shall see that
Vi 0 a Vil,
Call an open covering
U
of families of open sets open covering of
[0,I]
Ho(pro- SS),
as required.
of X x [0,i]
U x Va
where
depending upon
U U
a stacked covering if
U
is an open set in
and V a
X
is a union is an
(see [E- S]).
~'//,,~ X
Let U be a covering of
X • [0,i]
U = {Us x V IU~
by basic open sets, i.e.,
open in
X,
V~
open in
[0,I]}.
Such coverings are clearly cofinal in the inverse system of all coverings of X x [0,i]. U
For each
x
in X,
of x x [0,1] c X x [0,I].
consider the induced covering Because
[0,i]
is compact,
U
x
admits a finite X
n X
subcover, say
{Ux, i x Vx,ili = l ' 2 ' " " n x } "
the stacked covering
Let
Ux = i=la U ,i,
and form
256
u'
=
Clearly each open set of
{u x v x , i l i
U'
= 1,2,-",n
K,
let
U
~ ~
x}
is contained in an open set of
coverings are coflnal in all coverings of
To define
,
X x [0,i].
be a stacked covering of
(8.3.3)
U=
uu ~ U , { u
•
X x [0,i],
U'
is an open covering of
X.
Ku:RVN(U')
to
RVi 0
RVi I
• [0,i]
>RVN(U),
form the r e q u i r ~
with
= ((x0,t),(Xl,t),'-',(Xn,t))
homotopy
K = {Ku:RVN(U' ) x [0,i]
> RVN(U) IU
a stacked covering of
in
p r o - Top
K I = RVII:RVX
(8.3.4)
with
~ RV(X x [O,i]).
Proposition.
Then the induced map
Proof. appropriate
K 0 = RVio:RVX
Let
VA--->VX
> RV(X x [0,i])
SS J
X,
U'
as in (8.3.3)}
and
[]
A
be a closed subset of a topological is a cofibration
We may represent the map level category
[0,i]}
Then the homotopies
Ku((x0,Xl,-.-,Xn),t)
from
say
valfV~}
is an open covering of
where
Hence, stacked
U .
VA--->VX
as follows.
in
space
X.
pro - SS.
as a levelwise cofihration
Each open covering
U
of X
in an
257
induces an open covering
UIA
of A,
(8.3.5)
namely
UIA = {U n AIU E U },
and an i n c l u s i o n o f V i e t o r i s nerves
w(u IA) Because each open covering of adjoining the open set X
to A,
A.
{U]A}
A
> VN(U).
can be extended to an open covering of
X \ A,
X
by
the set of restrictions of open coverings of
(see (8.3.5)), is cofinal in the set of all open coverings of
We obtain the required representation
VA ! {VN(UIA) IU
(8.3.6)
VX/VA
Proposition.
an open covering of
Given
A c X,
X} ~
> {VN } E VX.
D
there is a natural map
> V(X/A).
Proof.
In the solid-arrow diagram
VA
> VX
,,,,
>VX/VA i
l +
V(X/A),
the composite mapping map.
VA-->VX--->V(X
/A)
is trivial.
This yields the required
D
(8.3.7)
Proposition.
Let
A
Then there is a natural equivalence sequence
VA--->VX--+V(X/A)
be a closed subset of a compact metric space VX/VA
> V(X/A)
in
is a cofibration sequence in
Ho(pro - SS), Ho(pro -SS).
X.
hence the
258
The following lemma about "shape cofibrations" is a key tool in the proof. is analogous to the statement that a map N
retract of
We state it in somewhat greater generality than is needed
[St -3].
A
in X
is a cofibration if and only if
there is a neighborhood N
of
A--->X
It
such that
now; we use the extra generality in Proposition
(8.3.8) let
U
Lemma.
Let
A
refines
N
is a strong deformation
(8.3.22), below.
be a closed subset of a compact metric space
be an open covering of
and a neighborhood
A
of A
X.
Then there is an open covering
in X
a)
V
b)
For each open set
c)
For each neighborhood
V
X,
and
of X
with the following properties:
U ; V
of V, N'
either
of A
V o A = ~
in N,
or
V 9 N;
the inclusion of Vietoris
nerves
VN(V]A) ---~W(VIN') is an equivalence in
Proof of Le~na. open cover, indexed by
in X
X
is compact, we may assume that
U = {Uili = 1,2,'--,n}. i,
UIA ~ {U i n A}. U'i
Because
Ho(SS).
i
ranges over
{l,2,o--,n}.
U i n A = U' i n A,
Conslder the restriction
[Kur, p. 122], there are open sets and if
v
Ui n A ~
> U' i
is a finite
From now on, in all constructions and sets
By a result of Kuratowski
such that
U
induce an isomorphism on Cech nerve~
U' E {U'.}I '
the inclusions
259 To do this, let
(8.3.9)
U'.l = {xld(x'Ui n A) < d ( x , A \ U i ) } ,
Next, perform the following constructions. U" = {U"i} , set with
let
N" = u U"i,
A c N c N r B".
(8.3.10)
Let
U"i = U i N U'i,
and, using normality of
X,
let
let
N
be an open
Next, let
V. 1 = U". n N, i 1 V. 2 = U". n (N" i i
A),
and
V. 3 = U. n ( X \ N ) , i l
N" \ \
N "'--..,
) -.
and let
VI = {vil},
_
V 2 = {Vi2},
Vi 2
~
and
V 3 = {Vi3}.
v" 3
Finally, let
V = V1 u V2 u V3.
Clearly our construction yields an open covering of Because the open sets are disjoint from
A
vll
are subsets of
N,
X
which refines
and the open sets
by construction, property (b) holds.
Vi2
U .
and V. 3 i
260
To check property (c), first note that the open sets
N,
hence for any open set
N'
with
(8.3.11)
3
are disjoint from
A c N' c N,
VIN' = (V 1 u V2)IN'.
Next, observe that each open set of
Vi
VIIN' ,
namely,
(8.3.12)
Vi2 n N'
vil N N'.
of V21N'
is contained in an open set
Hence,
VN(VIIN') = VN((V 1 u V 2) IN') = VN(VIN'),
and similarly,
(8.3.13)
Finally,
VN(VIIA) = VN(V 1 u V2) IA) = VN(VIA).
VIA = VIIA = VIA,
applied within
N'
and a look at Kuratowski's construction (8.3.9)
yields
(8.3.14) CN(VIA ) = CN(VIIA ) = CN(UIA ) ~ CN(U'IN ) a CN(VIIN) = CN(VIN),
via the inclusion
A c N.
the desired equivalence
(8.3.15) X V2
and V
VN(VIA) = VN(VIN) ,
Proof of Proposition
any open covering of
and V 3
Because
By Dowker's Theorem (8.2.10), formula (8.3.14) yields
N
as above, let
V'
is a neighborhood of
are disjoint from
A,
(8.3.7).
X
as required in (c).
Let
U
D
be an open covering of
constructed by Lemma (8.3.8) above.
be the open covering A
the projection
(in
X)
V 2 u V 3 u {N}
and the open sets of
~:X--->X/A
With
of X. V2
and V 3
induces an open covering
261
V" E wV'
of
X/A.
Because
V refines
V',
we obtain a co=m~utatlve diagram
VN(VIA) ~v~(v) . vN(v)/w(VIA) \\ i
(8.3.16)
,~.~ ~
bp
vt:cv"). We shall show that
p
is a weak equivalence.
Form the following commutative diagram.
VN(VJA) '"
)VN(V)
> VN(V)/VN(V IA)
q, (--)
(--) W(V]~)
,
w(v)
9 nICV)IVNCVlm
q (--) v.(v' I")
. vNCV')lW(V'
9 VN(V')
IN)
q"l(-')
(--3
t
v.(v' ]A)
p'
9 vs(V')IW(V'
, v~(V')
\
-,...
P"](~-) VN(V")
(8.3.17)
]A)
262
In diagram (8.3.17) the composite map
p"p'
and the rows are cofibration sequences in equivalences
a)
(or isomorphisms)
RVN(V'IN) = RVN({N}) (8.2.10)) = * the maps
so
is the map SS.
(because
The indicated maps are weak
N e V') = CN({N}) similarly,
~VN(V')/VN(V'IN)
weak equivalences.
in diagram (8.3.16),
for the following reasons.
VN(V'IN) = *
VN(V')
p
Hence
q"
and
(by Dowker's Theorem
VN(V'[A) = *.
VN(V')
Therefore
-->VN(V')/VN(V'IA)
is a weak equivalence
(by Axiom M5
for
SS).
b)
The map of
VN(VIA) ----->VN(VIN)
V and N.
equivalence
is a weak equivalence by the construction
Hence, the induced map of the cofibres,
(by [Q -i, Prop. 1.3.5] for
SS,
q',
is a weak
compare Proposition
(3.4.12)(c)).
c)
To show that
p"
is an isomorphism in
V' = {V 2} u {V3} from point
A
and a
N
N
recall that
where the open sets of
is a neighborhood
in A .
SS,
of
A,
V 2 and V 3
and
V" = ~,V'.
Then the required inverse of
p"
formula
p"-l(y0,Yl,-..,yn)
= (x0,xl,-..,Xn)
where
l a xi =
if
Yi = [A] s X/A
if
Yi e ~ ( X \ A )
-1 Yi
c X/A.
are disjoint Choose a
is given by the
are
263
d)
To show that
q
is a weak equivalence observe that
(8.3.18)
VN(V') : VN(V) UVN(VIN)VN({N})
(and recall that VN({N})
N E V'
so that
VN(V'IN) = VN({N})).
is contractible, the diagram of geometric realizations
RVN(VIN)
-> RVN({N})
1"
(8.3.19)
RVN(VIN ) -.
commutes up to homotopy (C
>CRVN(VIN )
denotes the unreduced cone).
the homotopy extension property to find a map
RVN({N})
which makes diagram (8.3.19) strictly commute. RVN(V')
Because
> RVN(V) uRVN(VIN)(RVN(VIN )
We may use > (RVN(VIN)
This yields a map
( R preserves eofibrations,
cones, and quotients by [Mil -2]) which extends the identity map of RVN(V)
and is a weak equivalence by an easy argument involving the
homotopy extension property. (see (a), above), the map
q
Because
VN(V') =
VN(V')/VN(V'IN)
of diagram (8.3.17) is a weak equivalence,
as required.
It follows that the map
p
in diagram (8.3.16) is a weak equivalence.
easy to check that open coverings of cofinal in all open coverings of
X/A.
X/A
of the form
V"
Therefore, the map
factors as a level weak equivalence (use the maps
p
It is
constructed above are VX/VA -
> V(x/A)
of diagrams (8.3.8)) followed
264
by a eofinal inclusion
{VNCV") } = V(S/A).
Thus
VX/VA -" V(X/A)
as
required.
(8.3.20) E VX
Proposition.
9V Z X
in
There are natural weak equivalences
Ho(pro -Top),
where
E
denotes the appropriate
unreduced
suspension.
Proof.
We use geometric realizations
(8.3.1), we may define a map the Vietoris complex of of
X)
X
as in Proposition
CRVX----~RVCX
(8.3.1).
Following
(from the cone of the realization of
to the realization of the Vietoris complex of the cone
which yields a commutative
solid arrow diagram
RVX
+ CRVX
> CRVX/RVX I I q
(8.3.21)
I I i
RVX
> RVCX
RVCX/RVX
in which the rows are cofibration sequences and the vertical maps are equivalences in
Ho(pro -Top).
CRVX/RVX But
> RVX/RVCX
Proposition in diagram
CRVX/RVX E E RVX a R E VX
RVCX~RVX = R(VCX/VX) Proposition follows.
(8.3.7) D
(R
(3.4.12) yields a filler (in
(8.3.21) which is also an equivalence (R
there.
preserves suspensions by [Mil - 2]) and
preserves quotients by [Mil - 2])
a RV Z X .
Ho(pro -Top))
Naturality
= RV(CX/X)
is easy to check.
(by
The conclusion
265
(8.3.22)
Propositlon.
spaces, and let
{X.} 3
X = limj{Xj}.
natural equivalence
Proof.
Let
VX
~ {VX.} 3
be an inverse system of compact metric
Then the projections in
X---> Xj
induce a
Ho(pro - SS).
By first applying the Marde~ic trick (Theorem ~.i.6)) if necessary,
we may assume that the indexing category
J = {j}
is a coflnite strongly
directed set.
Let
{Uj,klk~ Kj }
be the inverse system of all finite open coverings of
where the k - index is assigned so that for
U.],k
to
assume that each indexing category
Because each
Vietorls system of
ujs j{j}xKj
(8.3.23)
Uj,,k
is the pullback of
Again, by applying the Marde~ic trick if necessary, we may
Xj',k"
directed set.
j' > j,
Xj,
5,
where
j E J,
X.3 is compact,
hence isomorphic to it.
as follows:
(j,k) ~ (j',k')
is a cofinite strongly
__{VN(Uj,k)}
if
is cofinal in the
Assign a partial order to
j ~j'
and
k ~k'.
Then
limj{VXj} -= limj{{VN(Uj,k)}k~K } = {VN(Uj,k)}j c j, kE K] 3
(an inverse system is its own limit in any pro- category).
otherwise stated,
(J,k)
ranges over
We shall now write the natural map
From now on, unless
uj ~ j{j} • Kj.
VX---->limj VXj
as a composite of
several maps which will be later shown to be weak equivalences. X'. 3
Xj,
denote the image of
X
in X.. 3
To do this, let
Apply Leana (8.3.8) inductively to obtain
286
open coverings
Vj,k of Xj
and neighborhoods
Nj, k of X'.3 in Xj
with the
following properties: a)
Vj,k refines
{~j,k}k~Kj
for j ~ J;
b)
Vj,k refines
Uj,k,
c)
Vj, k refines
Vj,,k
{vj,k} d)
yielding inverse systems
Vj,k, for k' < k,
so
that
for j' < J,
whi~ is cofinal in
N.3,k c Nj,k,
{Vj,k}keK. 3
for k' < k,
is cofinal in
yielding an inverse system
{uj~,k} by (b); yielding inverse systems
for
3 e) Nj, k
is contained in the pullback of
yielding an inverse system
f) The inclusions in
Ho (SS),
N.3, k
to Xj, k
for j' < j
{VN~Vj,kIS,k)};
VN(Vj,klX'j)-r > VN(Vj,klNj ,k )
are equivalences
hence the levelwise inclusion
{VN(Vj,klX'j) } ' > {VN(Vj,klNj,k) }
Factor the natural map
VX ~
llmj{VXj}
is an equivalence in
as follows:
Ho(pro-SS).
267
(8.3.24)
~,- {w{uj,~]~j)}
VX
--" {wIs,~[~}
(by ( c ) ,
=> {v~5,ki~j,k)}
above)
(by (f), above)
i ) {VN(Vj,k)}
-
li%{{v~(Vj,k)}k~K.} .]
lim.{VX.} 3
(by (b), above).
3
We shall complete the proof by observing that the maps isomorphisms.
For
~
X~,
in Xj
for
For
j' > j
is contained in the open set the image of Vj, k
and i
above are pro-
this is an easy consequence of compactness and properties
of the product topology. of
~
Xj, in Xj
i,
consider any fixed index
(j,k).
The images
form a family of compact sets whose intersection Nj, k.
Hence we may choose a
is contained within
N.3,k.
j' > j
Because
w(vj,~l~j,k) ~ 'w(~j,,k)
Hence,
i
i s a pro - i s o m o r r h i s m , a s r e q u i r e d ,
such that
Vj,,k
(property (c), above), we obtain a commutative diagram
the conclusion follow&
X'j
refines
268
w
Proofs of Theorems
(8.4.1)
(8.2.19),
(8.2.20), and (8.2.21).
Proof Of Theorem (8.2.19).
(Proposition
Because
(8.3.1)) and the strong homology
homotopy invariant (5.6.7), the composites
preserves homotopies
V
Sh,
theory Sh
oV
on pro- SS
are homos
is
invariant.
n
Similarly, because exactness of
V
preserves cofibration sequences
Sh ~ V n
(8.2.3)(E) holds.
Sh n ~ V
follows from exactness Of
Let
RCN {Un+ I}
(by (5.6.7))
m
(by (5.6.7) and Proposition
>
Sh
n+l
{RCN {U }} n ~ RCN {U } n
{RVN {U }} = {RCN {U }} n n in
n
isomorphism
(8.4.2)
V o
> X
{VN(%)}
Let
(8.3.20)).
be a finite complex. Un
be the tower of Cech nerves. to rigidify in
tow- Ho(Top).
Then
VX
is a finite open covering of Choose bonding maps By [Dow], (see (8.2.10),
{RCN {U }}. n
But, because
X
is a complex,
But, by (5.2.13), the composite mapping
is an isomorphism in
~:Sh,--->h,
X
where each
Ho(tow- Top).
on finite complexes.
Proof Of Theorem (8.2.21).
of maps of spectra (see [Adams- i] or w our formula.
Thus, Axiom
Q
tow-Ho(Top).
n
{RVN {U )}
o
Proof of Theorem (8.2.20).
{RCN {U }} m X
(5.6.7).
= > Shn+ I o [ o V
admits a cofinal subtower X.
n
(8.3.7)),
For Axiom (S), the required natural equivalences are given by
The conclusion follows.
(8.4.2)
Sh
(Proposition
This yields a natural
D
Interpret products and operations in terms The conclusion follows easily from
269
w
Spectral sequences. We develop Bousfield-Kan and Atiyah-Hirzebrueh type spectral sequences which
converge to
Sh,.
Ken Brown [Brown] developed similar spectral sequences for
sheaf cohomology;
Kaminker and Schochet [K - S, Theorem (3.10)] obtained the
second spectral sequence using fundamental complexes. Ken spectral sequence to verify that pletes the proof that
(8.5.1)
Sh,
Theorem.
Sh,
We shall use the Bousfield-
satisfies Axiom (8.2.3(W)~
This com-
is a generalized (reduced) Steenrod homology theory.
(Bousfield-Kan spectral sequence).
Let
{X.}
be an
3
inverse system of compact metric spaces and let
X = lim {Xj}.
Then there is a
spectral sequence with
EP,q = limPj{Shq(Xj)}, 2
which converges completely under suitable circumstances to
Proof.
"Recall that
VX ~ {VXj}
spectral sequence (4.9.4) for "pro - (pro - SS)".
for
limP.{Shq(Xj)}j = 0
r > n +I
Now apply the Bousfield-Kan
applied to the inverse system
Xj
unless
the differentials
end at a 0 -group.
case.
pro- SS.
Compare Proposition (5.6.8).
In particular, suppose that
that
Shq
in
Sh,(X).
d
{VXj}
in
The conclusion follows.
D
is an inverse system of cardinality
=nH<
0 =< p =< n + l .
because
r
(of bidegree
This is complete convergence.
Then
Ep'q = EP~ q n+2
(r, r - i))
either begin or
We cite an important special
so
270
(89149
Corollary.
metric spaces and
0
X = lim {Xj}.
9 liml'3 {Shq+I(Xj)}
(8.5.3) Corollary.
Proof.
(Compare (4.9.3)).
Let
Then there are short exact sequences9
>Shq(X)
9 llmj {Shq(Xj)}
For compact metric spaces
~2''"
liraN { v
J =i
Apply Corollary (8.5.2) to this tower. Shq+i(Vj=l Xj) ---->Shq (Vj=l Xj)
Sh,.
|
Xj} = v
j =i
limI
D
consider the tower
'
N
xj IN = 1,2,--'} with
surjections, the
>0.
The strong wedge axiom ((8.3.3)(W)) holds for
N
{v
{Xj} be a tower of compact
Xj,
the stron 8 ~edge.
j =l
Because bonding maps
in the towers
{Shq+l(Vj=l Xj
terms vanish in this case.
are clearly
The conclusion follows.
D
(8.5.4) Remarks. a)
This completes the proof of Theorem (8.2.21).
b)
If a compact metric space of a tower of polyhedra
X
is represented as the limit
{Xj},
Corollary (8.5.2) yields
short exact sequences 0 9 > liml'3 {hq+l(X3 )}
relating
Sh, to h,.
Sh (X) q
~ limj {hq(Xj)}
>0
Compare Milnor's characterization of ordinary
(reduced) Steenrod homology [Mil -i].
Uniqueness does not follow in our
case because of possible extension problems; however, any natural transformation of 8teenrod extensions of h,
is an isomorphism by the above
271
short exact sequences.
(8.5.5)
Theorem.
(Atlyah-Hirzebruch
a compact metric space of dimension
spectral sequences)
d < ~.
[K- S].
Let
X
be
Then there is a spectral sequence
with E2p,q = SHp(X;hq(S0))
and
dr
of bidegree
(-r, r - i)
Sh,(X )
which converges to
in the sense that
E d+l = E~"
Our proof is contained in (8.5.6) -(8.5.21), below. X
of dimension
d < ~
there is a cofinal tower
For a compact metric space
{U }
in the Vietoris system
n
VX
such that each
U
is a finite open cover (use compactness) with
n
dim CN(U n) ~ d
(use the definition of covering dimension).
We therefore begin by
proving the following.
(8.5.6)
Theorem.
Let
X = {X
}
be
a tOWer of finite simplicial
complexes
n
and simplicial maps.
Then there is a spectral sequence with
E2p,q = SHp(X;hq(SO))
which converges dim X ~ d < ~ ,
Proof.
to
Sh,(X)
if
dim X
is finite.
More precisely,
if
Ed+l = E .
The proof is broken up into several steps:
(8.5.7) -(8.5.20), below.
272
(8.5.7)
(Compare
Construction of the spectral sequence.
FpX - {FpXn}
be the levelwise p- skeleton of
[K- S]).
Let
Following Massey, we define
X.
an exact couple
i
D I.
i
\/ i
where E1 = ..Shp+q(FpX/Fp- 1 X ) p,q
D lp,q = Shp+q(FpX),
degree
i ffi (i,-i),
degree
yields a spectral sequence
(8.5.9)
J ffi (0,0),
,
and
and degree
k ffi (-i,0).
This
{E;,q(X)}.
Description of
E 1 and
E 2.
For each
p and n,
(FpX/Fp_iX)n = (FpXnlFp_lX) = v Sp,
a finite wedge of p - spheres, one for each p -cell of
X
n
Hence~
E~,q = Shp+q{(VSP)n}.
A L s o , . b e c a u s e t h e b o n d i n g maps
Sp
(where
i
Xn - ~ X l
i> ( v SP)n+l
is a typical injection and
w
are simplicial,
> (v SP)n
t h e c o m p o s i t e maps
= ~ Sp
a typical projection) have degree 0 except
273
for at most one Sh,(S0)
~
for each
i;
in which case the degree is
+i.
Hence, if
is a graded ring, we may choose bases for the free modules
Shp+,( v SP)n -= (9 finite ~Sh*(S0))
(8.5.10) Shp+,((V SP)n+I)
so that the maps
~ Shp+,((v SP)n )
are represented by matrices of
the form
(8.5 .ll)
\
o
.
Hence, the towers
(8.5.12)
{ h p + q + l (vSp)nln ~ 0 }
are M_ittag-Leffler, so that
(8.5.13)
for all
p,q.
liml{hp +q +i ( v SP)n } = 0
(If
Sh,(S0)
is not a ring, analogues of (8.5.10) -(8.5.11)
still hold, but are more difficult to describe. (8.5.13) yields the following useful calculation.
Thus, (8.5.13) holds in general.)
274
(8.5.14)
Shp..~ { ( v SP)n} = hp+q{ ( v SP)n} = limn{hp+q( v SP)n}
= li.
({.p( ~ sP)n}; hq(S0))
= ~p({(~sP)n}; hq(S0)) = SHp ({(
Now, for each fixed with the
q = q0
v SP)n} ; hq(SO))
consider the exact couple (8.5.8) associated
generalized Steenrod homology theory
(8.5.15)
Skn(-) SHn_q0(,hq0 =
-
(sO))
~h
(so)
.=qo, \)
Skn(S0 ) = J q 0
In this case, the resulting
Ei = 0 P,q
n
unless
#
qo"
q = qn, %2
so that in the result-
ing spectral sequence
= Sfip(X; hq(SO)), E2p,q(X) =
E'p,q(X) = I 0Skp+q(X) otherwise.
This shows that
275
E2 (X) m SHp(X; hq(S0)), P,q
(8.5.16)
in the original spectral sequence, as required.
(8.5.17)
0
Convergence.
q ~ d,
so that
(8.5.18)
If
dim X = d < ~,
dd+ 2 = dd+ 3 = "'" = 0
Naturalit~.
in
unless
E
E d+l P,q
P,q
X--->Y
of towers of
Then there is a diagram
• ~Z<
Tow- SS,
and
E1 = 0 P,q
Consider a weak equivalence
simpllcial sets of bounded dimension.
(8.5.19)
then
Y,
hence an isomorphism
E~,q(X) a Sfip(X; hq(S0))
(8.5.20)
~>SHp(Z, ~ .
hq(SO))
<----SHp(Y, a ~ 9 hq(S0) )
E 2 (y). P,q
It is easy to see that the isomorphism (8.5.20) is independent of in (8.5.19).
Hence naturality follows from naturality of
Z
and the maps
S~,(;h,(80)).
This
concludes the proof of Theorem (8.5.6).
(8.5.21) sion
d,
that each
.Proof of Theorem (8.5.5).
With
X
a compact metric space of dimen-
choose a cofinal tower of open coverings of U n
is finite and satisfies
X,
dim CN(U ) ~ d . n
{Un},
in
V(X)
such
Next, choose bonding
276
maps to rigidify the Cech system
By Dowker [Dow], see (8.2.10),
CX E {CN(U )}. n
RVX = RCX
in
Hence, by (5.2.9),
tow-Ho(Top).
RVX = RCX
in
Ho(tow- Top).
Applying Theorem (8.5.6) to
RCX = (RCN(Un)}
required spectral sequence.
Naturallty follows from naturallty of the Vietorls
construction; replace
w
RCX
by
RVX
with
X = llmn{RCN(Un )}
in the formula for
E ,,,(x). 2
yields the
[]
Duality. We shall prove a Steenrod Duality Theorem (8.6.1) for our generalized Steenrod
homology theories and a more general theorem (8.6.2) involving functional duality. See w
(2.2.36) - (2.2.43) for Alexander and Spanier-Whltehead duality and
functional duality.
Kahn, Kaminker and Schochet [K - K -S] took the formula in
Theorem (8.6.2) as their definition of Steenrod homology; hence the two definitions agree up to (non-canonlcal) isomorphism.
Our present proof of (8.6.2) and its
application to (8.6.1) was motivated by a letter from Kaminker.
(8.6.1)
Theorem.
Let
X
he a compactum in
Sn.
Then
Sh (X) a hn-p-l(sn \X), P and the isomorphism is natural with respect to inclusion maps.
We shall first prove the following.
(8.6.2)
Theorem.
functional dual of
X.
Let
X
he a compact metric space, and let
Then
Sh (X) ~ h-P(Dx). P Theorem (8.6.1) will follow from (8.6.2).
DX
he the
277
Proof.
Let
E
be the spectrum which represents
inverse limit of finite complexes, CWSp
be the functional dual of
DXj ^ Xj
9 S0 .
X - lim {Xj}. X.. 3
h, . Let
Write
X
as an
DXj ~ HOM(Xj,SO)
in
Consider the evaluation maps
These induce maps
DX.j ^ Xj ^ E
~S O ^ E
9
hence by adjointness
DXj Xj A E ----~E
which are weak equivalences because
,
Xj = DDXj
for finite complexes.
Thus
Shp(X) = ~pS{Xj ^ E}
~pSIEDXj }
We therefore need to show that the natural map
hocolim {DXj }
(see w in a map
~DX
for homotopy colimits) is a weak equivalence.
~,sDX
represented by a stable map
X ^ Sn ~ S
O
or
X
Sn
First, consider a class
> HOM (X,SO) ,
>HOM (sn,s 0) -" S-n.
or by adJointness,
Because each map from
X
to
a polyhedron factors through a nerve (up to homotopy), the map w,S(hocolim {DXj}) ----~,sDX X x I --~Y,
Y
is an epimorphism.
Similarly, because each homotopy
a polyhedron, factors through a nerve of
=,S(hocolim {DXj}) ~
w,sDX
is a monomorphism.
X,
Therefore the natural map
278
hocolim {DXj}
is a weak equivalence,
> DX
so
s{Dxj}-~ ~ s P
{EDXj}
holim
P hocolim {DXj } __ ~ s E P (E
is stable)
-~ w SEDX P = h-P(Dx),
as required.
(8.6.3) (n- I) -dual
Q
Proof of Theorem (8.6.1). Sn \ X
the functional dual a sequence X.
{Xj}
DX.
were a polyhedron, (n- i) st
Theorem (8.6.1) would then follow.
of polyhedral neighborhoods {sn\x.} J
hence in
("dual" to (5.2.9)). Further,
X
would be stably equivalent to the
The direct systems
inj -Ho(Sp),
If
and
Ho(inj - Sp)
of
{En-~xj }
X
in S n
D
Instead we choose whose intersection is
by the comparison theorem for direct towers = h-P{Dx.}j
as required.
in Theorem (5.2.9) and its "dual" can be
carried out for pairs, so the isomorphism respect to inclusion maps.
suspension of
are then equivalent in
hn-p-l({s n \ Xj}) e hn-p-l{zn-~xj}
the required constructions
then the
Sh (X) e hn-p-l(s n \X) P
is natural with 1
w
SOME OPEN QUESTIONS
We shall give a brief list of some open questions.
(9.1)
For proving Whitehead Theorems and defining general profinite
comple-
tions, it would be useful to have a tractible description of coherent prohomotopy theory.
R. Vogt
[Vogt - i] gave a geometric description
coherently-homotopy-commutative between such diagrams.
of a category of
diagrams and coherent-homotopy
In [Por - 3], T. Porter has defined a coherent prohomotopy
theory eopro- Top, and shown that the natural functor from copro -Top
(9.2)
Ho(pro - Top)
to
is a natural equivalence.
Off towers,
is still mysterious. from an object of
(9.3)
classes of morphisms
If
the relationship between In particular,
does every object of
and
pro -Ho(Top)
p r o - Ho(Top)
come
pro - Top?
f:X~Y
~(f) E pro -Ho(Top)
Ho(pro - Top)
is a morphism of is invertible,
then is
Ho(pro- Top) f
Invertible?
such that This problem is
probably very delicat e .
(9.4)
Now that the homotopy
systematically obtained
study completions
to the usual one (e.g.,
theory of as
pro - SS
pro - objects
[B- K]).
is well de~eloped,
one should
and compare the theory thus
280
(9.5)
we expect applications of the machinery of these notes to the problem
of classifying open principal G-flbrations (G
a com~)act topological group) (see
[Coh]) and to the problem of defining the continuous algebraic K-groups of a topological ring (see [Wag]).
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J. F. Adams,
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[Adams - 2 ]
,
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Lectures on generalized cohomolgoy, in Category theory,
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[A - B]
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v]
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Index
Adams map 196 Artin-Mazur reindexing 6 Atiyah-Hirzebruch spectral seql~nce
271
basepoints in proper homotopy theory 225 Borsuk shape 229 Bousfield-Kan spectral sequence 166 (C, pro- C) 114 Ccf 47 Cech extension 251 C~apman complement theorem closed model category 41 cocylinder cofibrant 42 cofibration 41 in CJ 57 in pro- C 72 short ... sequence 97 trivial 42 cofinite 6 coherent pro -homotopy 124 comparison theorem 173 completions 181 condition N 45 CW (prespectrum, spectrum) cylinder 43 discrete 154 Dowker's theorem 251 duality Alexander 35 functional 36 Spanier-Whitehead Steenrod 276
230
13
35
Eilenberg-MacLane pro -space 203 Eile~berg-MaeLane spectrum 16 end 123, 216 of o -compact space 220 Ex~ 132 fibration 41 in C J 57 in pro - C 72 long ... sequence 104 short ... sequence 104 trivial 42 fibrant 42 filtered category (map, space, 35c.) filtering category 4 finite simplicial set 12 flasque 139
115
295
germ of proper map
214
hocolim 170 holim 130, 133 HOM 48, 108 HOMj 148 HOM 148 pro homology operations 38 see also pro-homology, homotopy idempotent 195 Hurewicz Theorem 187
inj-C
Steenrod homology, etc.
8
J -diagram K0 236 Kan complex
4
ii
lim (of a pro -abelian group) limS 139 loop -space 97
139
Maps (pro- C) 113 Mardew construction 6 Mittag-Leffler 162 strongly ... 163 model category 41 closed ... 41 simplicial closed ... 48 moveable 7 N
(condition N)
45
Postnikov system 187 pro- C 4 pro -homotopy groups 184 pro- homology groups 184 proper c a t e g o r y a t ~ 214 homotopy theory 214 section 222 weak . . . h o m o t o p y t h e o r y Q
233
reindexing 6 ring spectrum
37
S 233 Shape 226 category 226 theory 229 strong ... 231 Simple homotopy theory
235
218
296
simplicial closed model category 48 perspectra 27 sets i0 spectra 28 smash product of spectra 20 sphere spectrum 15 stable 7 stability problem 188, 191, 193 standard action 235 Steenrod extension 251 homology groups 247 homotopy groups 245 Strong homotopy groups 206 Strong homology groups 208 strongly directed 6 suspension 96 Tel 115 telescope 115 direct ... 169 trivial (cofibration, fibration)
42
V 250 van~shlnE theorem 139 vertical homotopy 221 Vletorls (construction, functor, etc.) W,W 155 weak equivalence
41
i n CJ 47 i n p r o - C 72 Wh 236 Whitehead Theorem 188, 193 countereza~ples 201 i n p r o p e r homotopy t h e o r y
- isomorphism
187
226
248, 250