TOPOLOGY and its APPLICATIONSTOPOLOGIJA i ee PRIMENENIJA
Budva 1972 (Beograd 1973) 200—212

PUPPE,DIETER,(Heidelberg, West

ON

THE

Germany)

STABLEHOMOTOPY

CATEGORY

1. HISTORY AND MOTIVATION. Many different attempts have been
made to find a good framework for stable homotopy theory. The best versions
so far were given in the semisimplicial setting by Kan [11], KAN and G. W.

WHITEHEAD [12] and in a geometric setting by BOARDMAN [2], [3], [19]. The
stable homotopy categories they construct are equivalent to each other (TIERNEY[17]). It seems that they have all good properties that one can reasonably expect,
but since in both approaches a lot of technical difficulties appear it is useful

to look for still other constructions of an equivalent
one

here

which

is also geometricnes some of Boardman’s
remarkbut

methods

in the introduction

more

with

to [3].

directcategory.

thanthose of [15].

RecentlyI

learntI shall describe

Boardman’s.

Itcombi-

idea

from a

I got

the

that

similaraapproach

was used by ADAMS in his Chicago lectures in 1971. The notes of these lectures [1] have just become available.
Before writing down the definition of myit. The first version of stable homotopy theory

category I shall try to motivate
was the S-theory of SPANIER

and J. H. C. WHITEHEAD [16]. It may be described as follows: Let FU be
the category whose objects are pointed spaces and whose morphism sets are

oF UW (4, B)=colim Tsp’ (Sk4, SB).
k—

Here

3

maps

and

oo

denotes reduced suspension, SJep®( , ) is the set of pointed continuous
the colimit (=
Tap?

(TX

directA,>limit)% B)—_—

is taken
Top?

over

(Zk+1

the suspension

4,

maps

> k+1B)

f=>Zf
There is an obvious notion of homotopy in 2

which gives rise to thehomotopy category Zh.
This category reflects nicely what happens in the“stable range” of ordinary homotopy theory. It is however not good enoughto allow all the constructions one would like to have. For several purposes(c.g.

Postnikov

systems

and

the

Adams

spectralsequence) one

shouldhave

THE

STABLE

objects.

If G

ON

Eilenberg-Mac

Mac

Lane

7

denotes

|

which

group

the stable homotopy

Eilenberg-

is a space B such that for all r&Z
|
r=n

H
Sp
mr (B)= 0, rn.

|

anand n&Z,

group

abelianLane object of type (G, n) in PZ
|

Here

is an

201

CATEGORYHOMOTOPYmay

be defined

|
by

|

Wh(S*% £98)SL
my (B)=
where

g is anythatr+¢>0.

Eilenberg-Mac

if K(G, n) is an

ordinary

Eilenberg-Mac

suchnumbernaturalare very rare in 70h. There are none of type (G, n) if G
summand which is finitely generated and not zero ([15] 8.3, 8.4).
On

the other

hand

Lane

objects

has a direct
Lane

space

of type (G, n) we do haveG, r=nw, K(G, =||r<2n-20, rn,

by the Freudenthal suspension theorem. Thus when n increases we got betterand better approximations to an Eilenberg-Mac Lane object. If K(G, n) is chosen to have the homotopy type of a CW-complex then there is a homotopy
equivalence K(G, n)— QK(G, n+ 1) and hence by adjointness of > and theloop space functor {2 a map> K(G, n)— K(G, n+ 1).
The sequence
Eilenberg-Mac

K(G, n)Lane

withtogether

these

maps

servesandobject in 70h

forms

a

as a model

forsubstituteanthe notion

of

for

a spectrum which was first considercd by Lima [13].
A spectrum X is a sequence of pointed spaces JX, (nEZ) together with given(pointed) maps &,: TX,— X,,,- Spectra are going to be the objects of our.
stable category. The difficulties begin when we want to define the morphisms
Y
spectrum
anothertoX
fromfmorphism
athat
The most naive definition is
that
such
Y,
f,:X,—
maps(pointed)ofis a sequence
|
£.
|
>

>

X,—X

n+l

J

|

| f, n+l

Nn

|
>

Y ntl

Y,—

name
commutes. But with this definition the category would not deserve theJAspectrum
the
4
space
pointed»otable™ It is natural to associate to any
defined by

w=

"4,
znd,
0

n>

n>0

n<0

a
(where a space consisting only of the base point is denoted by 0). Thenordinary
an
but
else
morphism JA— JB in the sense defined above is nothing
an element of(pointed) map 4— B. It should better be a stable” map, i.e.
for n>n,only
definedisf,thatallowingbyachievedbecan$2 (4, B). This
where n, may depend on f. Equivalently this means that a map of spectra

202

PUPPE, DIETER

X— Y needs only to be defined on a subspectrum X’ of X where X,=X,
for n large enough.
This is not yet the right notion, because one would not get a satisfactory
desuspension theorem (6.1, 6.2 below). For this theorem it is important thatthe domain of definition X' of f may still be smaller than stated above. A condition which one could use for this purpose is that for any » there is an r
such that X7., contains the image of ¥"X, under the map

bri STK, Ape, To>r-1g,

Sey

Sr,

&,

r—

But then there are other objections. The wedge of spectra should be a coproductin the stable homotopy category. This would not be true for
Vv JS,
r=0

because (using the definition of maps of spectra justproposed) J would induce

an embeddingof spectraof “Jpinto the homotopy

not a coproduct in 2hcategory

and

V\ S”

is

r=0

([15] 7.6).

A good condition for the domain of definition X’ of f seems to be:
1.1

For

any

n

and

any

compact(pointed)subsetKCX,

there existsanatural number r such that §,,. ,(Z'K) CX,yr.
One runs however into difficulties if the spaces involved are arbitrary.
Hence I restrict myself to CW-complexes and give now the systematic construction of the stable homotopy category”.
2. CONSTRUCTION. From now on we take all spaces, maps, subspaces
etc. to be pointed without repeating it always explicitly. A CW-complex shallhave a given cell decomposition and a O-cell as base point. The product 4 x Bof twoCW-complexes

A, B is takenwith the standard

decomposition

into pro-

duct cells and with the CW-topology (compactly generated topology). The smashproduct A AB is obtained from 4 x B as usual by identifying the two axes tothe base point andkeeping

all other cells.

S$” denotes

then-spheredecomposed

into the base point and one n-cell. The reduced suspension is defined by> A=S'ANA.
A spectrum X will now always be a sequence of CW-complexes X, together with CW-embeddings £,: > X,— X,,, (n&Z). This means that £, maps
> X,, isomorphically onto a subcomplex of X,,,. We call X, the terms and &,the structure mapsof X.

By .#» we denote the (preliminary) category of such spectra and ’‘naive”
maps of spectra. Thus a map f: X— Y in %p is a sequence of (pointed continuous) maps f,:X,— Y, such thatEn2>commutes(§,, 7, structure maps,

/ n

X, nT
|

X n+1
Lf n+1DALE an< Z).

ON
THE STABLE HOMOTOPY CATEGORY
A

spectrumX’is called a subspectrum of X if X,

is

a

~~

203

subcomplexof X,

for every n and the sequence of inclusions i,:X,CX, is a map i in Fp. X’~ is called dense in X and i is called a dense inclusion if condition 1.1 is satisfied
which we reformulate in the following equivalent form:
2.1

For anyncZ

number r suchand any finite

subcomplex

K

of X,

there

existsanatural_¥pby in-

that X,., contains the image of 3." K under

Ze,
Cntr:

SX,

Z'E,

En srmerly,

The stable category .

’ SALASwhich we propose is obtained

GN

from

verting all dense inclusions and making .% universal with respect to this property. More precisely: We look for a category .% together with a functor
P:.%p— 5 such that:

2.2 (a) Pi

is an(b) If P':.Pp—.F
functor F such thatisomorphismin ¥for anydense

inclusion i in Fp.

is any functor satisfying (a) then there is one and only one

P
JP, |
pu

P’

FE

commutes.
In the terminologyof GABRIEL-ZISMAN[5] p. 6.7 &°fractions of .%#p for the class of dense inclusions.

is

the

categoryof

P is necessarily bijective on

objects. Hence we may assume that .%° has the same objects as Sp and P is
the identity on objects. That .% exists is clear from general considerations [5].
But here the description of .%° is particularly simple because the class of dense
inclusions

in ,%padmitsa calculusof right fractionsin the sense of [5] p. 12.

The following properties are sufficient for this:
2.3 (a) All identities of .¥p

are dense inclusions.

(b) Any

inclusions

compositionof dense

is a dense inclusion.

(c) For each diagram X Ei Y L Y' in Fp, where j is a dense inclusion, there exists
a commutativesquaref

X—=Y]t I tJ.

X' —>Ysuch that i is a dense inclusion.
(d) Dense inclusions are monomorphisms|
in Fp.

Proof. (a), (b) and (d) are obvious. In (c) one may define X, as theunion of all subcomplexes L of X, such that f,L C im (J: Ya Y,).
Using [5] p. 12—14 it follows that a morphism X—Y in .% may be|

represented by a diagram

fo.

XOX’ > Y in ,#p whereX’'

is dense

in: X, and204

PUPPE,|DIETERg

Co.

:

that another such diagram X DX’ — Y represents the same morphism mm % if
and only if there exists a dense subspectrum X' of both X’ and X'' suchthat f and g coincide on X'”’. In other words #(X, Y) is the colimit ofFp (X', Y) where X’ runs through the dense subspectra of X.
3. COLIMITS.
our

We

shall

constructcoproducts

andcertainpushoutsin

category.

The

wedge

X=VX*
CA
of an arbitrary family of spectra X* is formed
corresponding terms and structure maps.

bytaking the wedge of the

3.1 Proposition. The wedge
coproduct in Fp and in SF.

thecanonicalof spectra

withinclusions is a

The proof is trivial for .#p. For ,% it relies on the fact that a subspectrum X’ of X is dense if and only if X' NX?

is dense in X* for all A.

The existence of pushouts is subject to analogous restrictions as in the
category of CW-complexes. A map f: X—Y in %p is called skeletal if f,: X,,— Y,
is skeletal for each » ie. f, maps each skeleton of X, into the corresponding
skeleton of Y,. fis called an embedding if f, is a CW-embedding for each n.
A map X—Y in % is called skeletal or an embedding resp. if it can be re-

|

Jo.

|

presented by a diagram XD X’'—Y in %p where X’ is dense in X and f has
the corresponding properties.
||
As an auxiliary notion we call an embedding f in %p closed if|i|> X,—X,

SHE,

:

1

a

>Y,—7Y,is a pullback

for eachn. Roughly

speaking

this

means

that

if thesuspensionof a cell e of Y, lies in (the image of) X then e itself lies in X,.

if

3.2 Proposition. Let Z«~—X—Y
an embedding

and f is skeletal.

be a diagram in Fp

In the case ,%the diagramout). In the case Fp the diagram has a colimit provided
The canonical functor Fp— 5° preserves these colimits.

Proof. Consider first the case Fp

or SF,

where i ishas a colimit(push-

that i is also closed.

and assume that § is closed. Then thecolimit may be constructed term by term in the obvious way. In the case ,%one has to look at suitable dense subspectra of the given ones. The hypothesis
closed” 1s not needed in this case because any embedding X— Z in %p
be factored into a closed embedding X— Z’ and a dense inclusion Z'CZ.

Another kind of colimits will be considered in section 8.

|

can

|

ON

THE

STABLEHOMOTOPY

CATEGORY

4. SMALL SMASH PRODUCTS. Let X be acomplex. We define the “small smash product” X AA

terms

205

spectrum and 4 a CWto be the spectrum with

X, A A and structure maps.
g, NidSX, NAD=S'ANX,NA—X,.
ANA

Obviously we get a functor
AoF P X UO ~

where
maps.

20° denotes the category
It induces a functor

5 P

of

(pointed)

oF XU

A
— F.

CW-complexesThis follows from easy general consideration about

categories

fromcontinuousof fractions and|If i: X'— X is a dense

4.1, Lemma.

:

and

X'NA—>

inclusion(up to a canonical isomorphism

XNA

in X N\A).

|

in Fp

and

its image

N

Proof. If K is a finite subcomplex

complex L of X, such thatKCL AA.

of X, AA

By

hypothesis

there

4.2 Proposition.

Let

A be a fixed CW-complex.

existsa finitesubthere is an r such tha

sr is mapped into X,,, by the structure maps, hence
mapped into Xi. , AAd=(X"ANA4),,,.
of Fp

then so is i Nid:

X' N\Abetweenis"KCS"ALAA

The endofunctors X +> X \ 4

and FP preserve coproducts and the pushouts considered in 3.2.

Proof. Since a coproduct is just a wedge (3.1) the first assertion is trivial.
the second one looks at the explicit construction of the pushouts in 3.2.
We shall also use smash product 4 AX where the CW-complex 4 is theleft factor. Everything is exactly the same as for X' A A except that the structure

For

maps

are given by|

SIAANY, TNE gp stax, BN

(4.3)

gp x

where «© interchanges the factors. A AX is canonically jsomorphic to XA A by
the map which interchanges the factors in each term.
In section 7 “large” smash products will be considered where both factors
are spectra.

5. HOMOTOPY. Let I be the unit interval considered as an unpointed

CW-complex with one l-cell and two O-cells. I+ is obtained by adding an
isolated basepoint. Let z be the nonbase-point of S° and define (pointed) maps|

Jo: SOT,
gq I[+—S°

v=0,1

|

206

by

PUPPE,

j.,(2)=v,¢ (I)={z}.

DIETER

This gives rise to cylinder functors X+>I+

AX in

Fp

and .% and to natural transformationsx=ax NE Lox
gr A XT NE gon xox
Together

Kamps

they

form

homotopy

systems

in

%p

and

,%

resp.

in the sence of[7], [8], [9].
5.1. Proposition.

Thehomotopy

systems justdefinedsatisfy

all extension

conditions DNE (n, v, k) (cf. [7](2.6), [81(3.3) or [9] (3.2).
ANI*=(I")*

wherethey are trivial and forms then the smash product with a spectrum.
Hence for ,*p and .% we get as corollaries all the results which

Proof.

KAMps

provedOne

constructs

for categories

the extensions

withhomotopyinsystem

I*AI*Asatisfying

In particular we have homotopy notions which aregive rise to the homotopy (quotient) categories pp
5.2 Proposition.

The categories Fp...

the extension

equivalenceand Fp.

conditions.

relationsand .%° have double mappingandcylinders

for all skeletal maps.

Proof.

Let

fr.

Y<« X —Y’be skeletal maps.
XV

< Jo»

yJo

J1>

i

INI

Then by 3.2 the pushout
A X

y

YNY'—————Zy, 5exists (because <j, j,>is a closed embedding).
cylinder.
By specializing we get:_(a) the mapping cylinder Z,=Zy, uy),(b) the mapping cone Cp=Z, ,, where 0:X—0
the zeroZ, s, is the

doublemapping

denotesunique

map

the

into

spectrum,(c) the suspension SX=C,=2Z
It is obvious from the definitions that
3.3 Proposition.

X

is canonically isomorphic

The cylinder functor in Fp

and &to SAX.

preserves all colimits

considered so far, namely
coproducts,
pushouts as in 3.2,
(double) mapping cylinders and mapping cones of skeletal maps,
Suspensions.

This is a corollary of 4.2 and the proof of 5.2.
The

resultsof 5.1

and

5.3

suffice to carryoutof a cofibre sequencef

X—Y->C,>TX—>3Y—>-

the standard

construction

ON

for all skeletal

maps

THE

maps

([7]

STABLE

section

HOMOTOPY

CATEGORY[10] section

5).

7,

The

207

restriction to skeletal

is not essential bacause we have.
5.4 Proposition. (a) If the spectrum X is bounded below (i.e there exists nycZ

such that X,=0 for n<n,) then any f& Pp
map.
(b) Any map(X, Y) is homotopic

to a skeletal

in 5° is homotopic to a skeletal map.

Proof. (a) Using the relative skeletal (cellular) approximation theorem for CW-complexes we deform f, into a skeletal map by induction over » making thedeformationcompatible

with(b) follows fromdiagram(a)f

X DX’ -—Y

the structuremaps.

becauseany

map in .%° may

where

X’

is dense in X and X’ is bounded below.

in %p

be

represented

by a

6. DESUSPENSION. To justify the name “stable” for our category wehave to show that the suspension functor is invertible up to equivalence. To
see this we introduce a new kind of suspension, the translation suspension, andcompare it to the other one which, for distinction, shall be called geometricsuspension.

The translationsuspension

one

step.

Thus

the spectrum

ture

maps

(Z:8n=E8ns1¥,

is simply

defined

3 ,X has the terms

byshifting(X,X),=X,,

the

, and

indicesthe struc-6.1 Theorem. The geometric suspension > and the translation suspension 3,
are naturally equivalent as endofunctors of Fp.
6.2 Corollary.
natural equivalence.

The geometric suspension

Proof.

invertible.

¥, is obviously

6.3 Corollary. ,%}Y :.Fh—Fb

is invertible up tois an additive category with arbitrary coproducts.

Proof. Up to equivalence every object of #4 is of the form YX andevery morphism of the form I f (6.2). From the usual cogroup structure on S*
one gets a cogroupstructureon>

X=S'AXandhence

a groupstructureonthe morphism sets of ,%’4 which is compatible with composition. The coproductin ,% (3.1) is also a coproduct in 5, because the cylinder functor preservescoproducts(5.3).

We mayturn 4

into a graded category 5
7b,

For

any

(X,

Y)=.%hskeletal map f we consider(25X,

the cofibref

X—Y—>C—and compose

the last

map

with

the

natural

X—->Y->C —X

has degree —1.

Y).

sequence|

2% X

obtaining a “triangle”
in which the last map

by defining

equivalence»X—3,X

of

6.1,

208

PUPPE,DIETER

6.4 Theorem. “ph, together with the class of triangles
triangle of the form just described is a triangulated category.
LER

isomorphic

to someThe name triangulated” 1s taken from VERDIER [18] (compare also HEL[6]). Almost the same concept was called stable’ in [14] and [15]. There

are minor differences In conventions about signs and there is an additional axiom in [18] not contained in [14] or [15], the octahedral axiom. What wemean here is that all the axioms are satisfied including the octahedral axiom.
We use however the sign conventions of [15]. The proof of the axioms is rather straightforward from the properties of the cofibre sequence given in [7]section 7. One may also take the proof of [15] Satz 3.5 and adapt it to thepresent situation with slight changes.
Proof of Theorem 6.1. Let

andstructure maps

3; X be the(3;&),= S£&,.

spectrum with terms(SZ) X,=

3 X,,

Then the maps

-,

En

|

(ZX), =ZX,~—X,|=(Z,X),

fit together forming an embedding £: 3;X— >, X in .%p The image is a dense
subspectrum of 3 ,X, hence £ is an isomorphism in ,%.At first sight one could
think that >:X equals 3X (which would finish the proof), but this is not so.
The structure maps of 3 X are

|

TAid

id NE,(28), STASTAX, ——S'ASIANX,——S'AX,,,
where © interchanges
the structure maps

the twofactorsS!

(4.3).

They

are not evenhomotopic to

id NE,(2:8),=,:STASIANX,

——S'

AX,

of +X. To get around this difficulty we consider the reflection ¢ of S? (givenby c(f)=1—1t if S1=7/{0, 1}) and define $_X to be the spectrum withterms (¥ _X),= XX, and structure maps
|

TA id

oc A&,(Z-8),:STASTANX, —=STAS'AX, —>S'
AX, ,,.
Now

we

have

an isomorphism|

k:SX—>S_X|
in Fp

given byk,=c"Nid: S*NX,—>S'
NX,

and on the other hand (¥ _%), is homotopic to (X:£),. One can show that
this latter fact implies that 3 _X is isomorphic to ¥;X in 5h. Actually wehave to prove a little more, namely that there is a natural isomorphism.
Forthis wechoosea specific (pointed)

homotopyof: SIAST—>SIASYte]|ON THE STABLE HOMOTOPYCATEGORY

209

such that p°=(c Aid)t and p!=id. Then

is a homotopy

EA id

SIASIAX,CL

fromto (X:8),.

(3X _%),(MX), =S'ANX,ANI"

andstructure

SAS

AX,

id AE,SL

Let MXSAX,

bethe spectrum with terms

maps(ME), :S'ANS'ANX, ANIT>S' NX,

ATT(S15 855 x, 1) > (ANE) (0° (sy, 8),

X), 1).

Define embeddings in
hy: > X—=>MX,

i:

X>MX

|

by i, ,(s, X)=(s, x, v), S&S, x&X,, v=0,1. Obviously M may be considered
as an endofunctor of ,’p and i, as a natural transformation. i, , is a homotopy equivalence for each single n. By Lemma 6.5 below, i, is an isomorphism

in #phprovided that the spectrum X is bounded below(compare 5.4). This

last restriction is irrelevant as soon as we pass to 5h, because any spectrum X
has a dense subspectrum X’ which is bounded below and X’ is isomorphic to X
in .%. Altogether we have

SX

——>
—

Fp

s x—— MYT T Fp——Feb

so x2—- SX .

TTFEach arrow is a natural equivalence in the category indicated (if we assume X
to be bounded below) and induces a natural equivalence in %h (without
restriction).
6.5. Lemma. Let f:Y—Y be a map in Fp such that f,:X,—~Y, is a
homotopy equivalence for each n and X, Y are bounded below. Then f is a homotopy equivalence in Fp, i.e. an isomorphism in pp.
Proof. We

shall show that for any spectrum

Je:

Z the map

pb (Z, X)—> Spb (Z, Y)

induced by f is a surjection. Taking Z=Y this gives a map g&.%p (¥, X)
such that fg is homotopic to id, In Sp. Since g, is a homotopy inverse of f,
we

may

apply

everything to g insteadof f obtaining

A<

%p

(X,

Y)

such thatgh~id,. This shows that g and hence f are isomorphisms in ph.
To prove f, surjective we assume X,=Y,=Z,=0 for n<0. This is noloss of generality and it simplifies the notation a little. Let y& %p (Z, Y)
Since f, is a homotopy equivalence there is a map x,:Z,—>X, such that
Voto Xx, Let ©0:Z, NI*—Y, be a homotopy deforming y, into f,x,. Using
the homotopy

extensionpy ¢°: ZAl*—Y
14 TOPOLOGY

propertyfor {,:> Z,—~Z,,,deforming y into some y°<andist APPLICATIONS,

Budva

we extend

7p (Z,

1972 (Beograd

1973)

Y).

0 to a homoto-

210

PUPPE,DIETER

Next we look at the commutative diagram

|

0

wl

0

¥

1s

Eo

No

|

|

¥

2 YyT> YY,

in which we would like to fill in x,:Z—X,.

£

EE

We consider Z,, X,, ¥,as spa-ces under YZ, by the maps %,, £ (Sx,) and vn, (yo) =F, &, (3 x,) resp. Then,
is a map under YZ, and an ordinary homotopy equivalence. Since {, is a cofibration it follows that f, induces a bijection of homotopy classes Z,—X, rel.
> Z, into homotopy classes Z —Y, rel ¥Z, ([4](10.5) Satz, p. 165). So wefind a map x,:Z~—X, and a homotopy ¢i:2Z, AI+—Y, deforming yi into f, x, .
We extend this to a homotopy ¢':Z AI*—Y deforming )° into some y'& Fp(Z, Y) such that © :ZyANI*—=Y, 18 constantwith respect to the homotopy
parameter.
Continuing in this way we recursively construct

Xp: Ly>X,eS pz ,Y)
"ES p (ZAI, Y)

such that|Son X= Yno" deforms

y"—!

into »”|or is constant with respect to thehomotopy

These maps fit together

parameter

giving XE

deforming y into fx, where|if k<n.

Fp (Z, X)

andoc

Pp (ZA[—-1,0]%,Pr (2,1) =or (z, 1),

n—-1<t<nfixe (2),t=o0|Hence f, [x]= [y].
7. LARGE
products

between|
SMASH
spectraPRODUCTS.
andIn section 4 we

CW-complexes.

have

It is importanttodefinedsmash

extendthis tosmash products of two spectra.
7.1. Theorem.

There

is a functor,

the ’’large smash product,A

Fb x Fb
with

TY)

oo

the following properties:

oo

Fb

|
|

ON THE STABLEY CATEGOR
CATEGORYY
TIOMOTOPY
5iABE HOMOTOP
cer

—_—

(a) It is associative and commutative
(b)

The

211
211

~~

up to natural equivalence.

diagramidx J|

Fb XU b—Fox Fb

CN

AN

Uh

/
yd A

|
| A

\
NN

xid

J

/

Fb

J

|

N |S

Jb

|

(which relates small and large smash products) commutes up to natural equivalence.
This implies easilyequivalence.

that JS°is a unit for the large smash(c) All the natural equivalences

product up to naturalin (a) and (b) can be chosen to be coherent.

The proof is too long to be included here. For Boardman’s stable category there
is a proof in [3] Chapt. II § 7. The proof in [19] has a gap in the proof of
Lemma 9.6 p. 154. A very direct proof of 7.1 for precisely the present situation
is given by
Boardman’s

Apawms [1] §4. My own proof is different from Adams’ but
ideas: For the purpose of this proof I pass from ordinary

follows
spectrato a new kind where the indexing set Z is replaced by the set € of finite dimensional subspaces of a euclidean vector space of countable dimension. The
structure maps have the form
S B, A N\ X 4X B>

where

4, BEE,

ACB

andSg 4, 1s the one point compactificationof the ortho

gonal complement of 4 in B. Details will appear elsewhere,
8. COMPARISON TO BOARDMAN’S CATEGORY. BoARDMAN defines
first the category .¥ , of finite spectra by stabilizing” the category .¥ of fiInite CW-complexes (cf. [3], [17], [19]). It is canonically isomorphic to the fullsubcategory¢ of .%

whose

objects havethe form

3: J4,

where

n€Z

and 4

is a finite CW-complex. The isomorphism sends 3; J4 into (4,—n) in the notation of [3] Chapt. II, 2.1 or [19] p. 81. It takes embeddings In ¢ Into embedings in .% , (which Boardman calls inclusions). Following Boardman we denotethe subcategory of embeddings always by putting an J in front of the symbol
for

whole

category.

BOARDMAN obtains his stable category .¥ ,, by applying his w-construction to the pair ((% ,, AF ,). Roughly this means that one adjoins to .7 ,,colimits of all directed diagrams in J,¥ ;,. We have

8.1. Proposition. Let & be Fp or F. Any directed diagram in JG
has a colimit in JE which is also a colimit in 7.
For lack
The proof is trivial for ¥ =.%p. It is more subtle for =.
of space we cannot give it here.

212
~s

PUPPE.

DIETER

By the basic properties of the w-construction
and only one functor G which makes(TF 6 JF 9

CT

TF wr

|
(©,

8.1 implies that there is one

LF sw)

v
FG)

CC

(L

G
JL)

commutative and preserves colimits of directed diagrams of embeddings.
2.5

TierNEY

has

given conditions

under

which

In [17]

G is an equivalence of categories.

It is easy to verify these conditions in the present situation.

References[1] J. F. Apawms, Stable homotopy and generalised homology, University of Chicago,
Mathematics Lecture Notes prepared by R. Ming 1971.
[2] J. M. BOARDMAN, Stable homotopy theory, mineographed notes, University of
Warwick 1963.
[3] J. M. BOARDMAN, Stable homotopy theory, mimeographed notes, The John Hopkins
University 1969/70.
[4] T. Tom Deck, K. H. Kamps and D. Puprpe, Homotopietheorie, Lecture Notes inMathematics 157, Springer 1970.
[5] P. GABRIEL and M. ZisMAN, Calculus of fractions and homotopy theory, Ergebnisseder Mathematik und ihrer Grenzgebiete 35, Springer 1967.
[6] A. HELLER. Stable homotopy categories, Bull. Am. Math. Soc. 74, 28-63 (1968),[71 XK. H. Kamps, Faserungen und Cofaserungen in Kategorien mit Homotopiesystem,
Dissertation,

Saarbriicken

1968.

[8] K. H. Kamps, Fibrations and cofibrations in categories with homotopy system. Proc.
Intern. Symp. Topol. and Appl. Herceg-Novi (Yugoslavia) 1968, 211-218 (1969).
[9] K. H. Kamps, Kan-Bedingungen und abstrakte Homotopietheorie, Math. Z. 124,
215-236 (1972).
[10] K. H. Kamps, On exact sequences in homotopy theory, Proc. Intern. Symp. Topol.
and Appl. Budva (Yugoslavia) 1972, 144-156[11] D. M. KAN, Semisimplicial spectra, Ill. J. Math. 7, 463-478 (1963).
[12] D. M. KaN and G.W. Whitehead, The reduced join of two spectra, Topology 3,
Suppl. 2, 239-261 (1965).
[13] E. L. LMA, Duality and Postnikov invariants, Thesis, University of Chicago 1958.
[14] D. PuppE, On the formal structure of stable homotopy theory, Colloq. on Algebraic
Topology Aarhus 1962, 65-71, mineographed.
[15] D. Purrg, Stabile Homotopietheorie 1, Math. Ann. 169, 243-274 (1967).
[16] E. H. Spanier and J H.C. WHITEHEAD, A first approximation to homotopy theory,
Proc. Nat. Acad Sci. U. S. A. 39, 655-660 (1953).
[17] M. Tierney, Categorical constuctions in stable homotopy theory, Lecture Notes inMathematics 87, Springer 1969.
[18] J. L. VERDIER, Catégories dérivées, Quelques résultats(Etat 0)IHES
Paris
1966, mimeographed.

[19] R. VoGT,
21 (1970).

(Received

16 10. 1972)

Boardman’s

stable homotopy

category,

AarhusDieter Puppe
Mathematisches
D

Lecture Notes

Series No.

Institut der Universitit

69 Heidelberg,

Im Neuenheimer Feld 9