On stable cohomotopy groups of compact spaces

l

es

paces.. They

are es-n thehen it

aperfof theas-s thatrespondexcep-

Topology and its Applications 153 (2005) 464–476

www.elsevier.com/locate/topo

On stable cohomotopy groups of compact spac

Sławomir Nowak

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Received 5 February 2003; accepted 13 October 2003

Abstract

The stable cohomotopy theory is a useful tool for studying global properties of compact sIn the stable shape theory we have infinite-dimensional versions of the Whitehead theoremrequire to use the stable cohomotopy groups. 2004 Elsevier B.V. All rights reserved.

MSC:primary 55P55, 55P40; secondary 55N05, 55Q07

Keywords:Stable cohomotopy groups; Stable shape

The purpose of the present note is to show that the stable cohomotopy groupspecially useful when we work on infinite-dimensional Hausdorff compact spaces. Oother hand we try to reduce the role of the stable cohomotopy theory to the cases wis irreplaceable.

It is known (see [11] and [18]) that the stable shape morphismf :X → Y is an isomor-phism if and only iff induces isomorphisms of all stable cohomotopy groups. In the pit is shown that (roughly speaking) it suffices to assume thatf induces isomorphisms othe stable cohomotopy groups in almost all positive dimensions and isomorphismscohomology groups with coefficients inZ. The last hypothesis can be replaced by thesumption thatf induces isomorphisms of the stable homotopy progroups. This meanin the stable shape category there are versions of the Whitehead theorem which corexactly to the Whitehead theorems in the classical shape theory with the only one

E-mail address:[email protected] (S. Nowak).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2003.10.012

S. Nowak / Topology and its Applications 153 (2005) 464–476 465

sitive

eoremiyata

basic

e CW

at

e

h

f

om-and a

tion. Instead of the conditions concerning finiteness of the shape dimension ofX andY weassume thatf induces isomorphisms of the stable cohomotopy groups in almost all podimensions orf induces isomorphisms of the stable homotopy progroups.

The last fact can be treated as a commentary to the version of the Whitehead thfor generalized stable shape category (Theorems 6.1 and 6.2 of [15]) proved by T. Mand J. Segal.

This assumption is essential because there exists [12, p. 153] a continuumX havingnontrivial stable cohomotopy groupsπn

s (X) �= 0 for infinitely manyn > 0 and trivialCechcohomology groupsHn(X;Z) = 0 for all n > 0.

1. Preliminaries

All spaces are pointed. The knowledge of the shape theory (see [2] and [12]) andnotions of the stable homotopy theory (see [1,21] or [13]) is assumed.

By S we denote the reduced suspension functor. The homotopy category of finitspectra is denoted bySCWf .

If E = {En} is a finite CW spectrum, then thenth termEn of E is a finite CW complexandEn+1 = S(En) for sufficiently largen. Without loss of generality we may assume tha morphismf ∈ SCWf (X,Y) is represented by a sequence of maps{fn}∞n>n0

(a map ofspectra), wherefn :Xn → Yn are defined forn � n0 and satisfy the conditionS(fn) =fn+1.

The sequences{fn}∞n�n1and {gn}∞n�n2

represent the same morphism ofSCWf ifffn � gn for sufficiently largen. We write [{fn}∞n�n1

] = [{gn}∞n�n2] and say that they ar

homotopic.This means that one can study properties of thenth term fn of the sequence whic

representsf instead of properties of the whole sequence.In particular, the reduced mapping cylinder off = [{fn}∞n>n0

∈ SCWf (X,Y) is definedasMf = {Mn}, whereMn = Mfn is the reduced mapping cylinder offn for n � n0 andMn = {point} for n < n0. We have alsoSm−n(fn) = fm andSm−n(Mfn) = MSm−n(fn) forn > n0.

Consider the commutative diagram of the categorySCWf

X1f1

r1

Y1

r2

X2f2

Y2

where all arrows are maps of spectra. Then there exists a map of spectrar :M1 → M2,whereMi = Mf i

denotes the mapping cylinder off i for i = 1,2. The restrictions ofr toX1 and Y1 (in a canonical wayXi and Yi can be identified with subspectra ofMi fori = 1,2) equal respectively tor1 andr2. The map representingr depends on the choice ohomotopy joining the maps which representr1 andr2.

More generally, if we prove a theorem in the homotopy category of finite CW cplexes in a controlled way (i.e., maps should commute with the reduced suspension

466 S. Nowak / Topology and its Applications 153 (2005) 464–476

(or the

remsts are

stable

.e

off the

yi-

e can

[1,

m

elian

gs(the

groups

rat

given mapf may be replaced by its suspension or then-fold suspensionSn(f )), then weget also a corresponding theorem for the homotopy category of finite CW spectraprohomotopy category of finite CW spectra).

For simplicity we adopt a similar convention in the sequel. More precisely, theoare formulated for the homotopy category of finite spectra, but proofs of auxiliary facpresented mostly for the category of CW complexes.

Analogously as in the case of the (ordinary) shape theory one can introduce theshape category (see [3,8,15,14,17] and [18]) of compact Hausdorff spaces.

By X andY we denote respectively the inverse limits of the inverse systems{Xσ ,pτσ }

and{Yς , qυς } of finite CW complexes in the category of topological spaces and maps

The morphisms of the stable shape category ShStab fromX to Y are represented by thelements of the Abelian group

Pro-SCWf (X,Y ) = lim←−ς lim−→σ SCWf

(Sus(Xσ ),Sus(Yς )

) ≈ ShStab(X,Y ),

whereX = {Xσ ,pτσ }, Y = {Yς ,qυ

ς } and byXσ ,Yς we denote the suspension spectraXσ ,Yς . By pτ

σ andqυς we denote the morphisms induced by the bounding maps o

inverse systems.Consider a reduced generalized cohomology theoryH = {hn} defined on the homotop

category of finite pointed CW complexesHCWf consisting of a family of contravarant functorshn :HCWf → AB with a family of natural equivalencesεn : hn+1 → hn · S,whereAB denotes the category of Abelian groups [21, p. 124]. In a canonical way wextendH = {hn} over the categorySCWf .

The stable cohomotopy groupsπns (−) form a generalized cohomology theory (see

13] or [21]) onSCWf .A generalized cohomology theoryH = {hn} defined on the categorySCWf can be

extended over Pro-SCWf . The grouphn(X ) equals to the direct limit of the syste{hn(Xσ ), hn(pτ

σ )}, whereX = {Xσ ,pτσ }.

This allows us do define a contravariant functor from ShStab to the category of Abgroups, i.e., theCech extension ofH = {hn}.

The Cech cohomology groupshn(X) of a compact Hausdorff spaceX are equal tothe direct limit of the system{hn(|N(α)|)}, whereα varies over the finite open coverinof X. The generalizedCech cohomology groups commute with limits of compactacontinuity property).

If X is a compact space, thenπns (X) is isomorphic to the direct limit of the sequence

{[X,Sn] Suspension−−−−−−→[S(X),Sn+1] Suspension−−−−−−→[

S2(X),Sn+2] Suspension−−−−−−→· · ·}.Together with the stable cohomotopy groups we consider the stable homotopy

πss (−). If P is a CW complex thenπs

n(P ) is the direct limit of the system

{[Sn,P ] Suspension−−−−−−→[Sn+1,S(P )

] Suspension−−−−−−→[Sn+2,S2(P )

] Suspension−−−−−−→· · ·}.This construction determines a covariant functor fromSCWf to AB.Applying this functor to the objectX = {Xσ ,pτ

σ } ∈ ObPro-SCWf we define a functoPro-πs

n(−) from the category Pro-SCWf to the category of progroups. It is clear thPro-πs

n(−) can be also defined on ShStab.


hap-

2. Maps between pairs of CW complexes

The following lemma was proved in [16, p. 251].

Lemma 2.1. Let (Xi,Ai) be a pair of finite connected CW complexes fori = 0,1, . . . , r

andfi : (Xi,Ai) → (Xi−1,Ai−1) be a cellular map fori = 1,2, . . . , r such thatπ1(X0) =0 = π1(A0) and πr+1−i (fi) :πr+1−i (Xi,Ai) → πr+1−i (Xi−1,Ai−1) is a trivial homor-phism for i = 1,2, . . . , r . Then there exists a mapg : (Xr,Ar) → (X0,A0) such thatf1f2 · · ·fr � g relAr andg(X

(r)r ∪ Ar) ⊂ A0.

Lemma 2.2. Suppose that(Xi,Ai) is a pair of finite CW complexes fori = 0,1, . . . , n

andfi : (Xi,Ai) → (Xi−1,Ai−1) is a cellular map fori = 1,2, . . . , n. Let r be a naturalnumberr � 1 satisfying the following conditions:

(1) X0 andA0 are r-connected.(2) The homorphisms

Hr+i (fi) :Hr+i (Xi−1,Ai−1;Z) → Hr+i (Xi,Ai;Z) and

Hr+i+1(fi) :Hr+i+1(Xi−1,Ai−1;Z) → Hr+i+1(Xi,Ai;Z)

are trivial for i = 1,2, . . . , n.

Then there existsg : (Xn,An) → (X0,A0) such that f1f2 · · ·fn � g relAn andg(X

(n+r)n ∪ An) ⊂ A0.

Proof. Suppose thatK is a simplicial complex andL is a subcomplex ofK andf : (|K|, |L|) → (Y,B) is a map such thatf (|K(n−1)| ∪ |L|) ⊂ B, where (Y,B) is a pairof simply connected CW complexes. S.T. Hu (see [9, p. 204] or Exercise E-7 from Cter VI of [10]) assigned tof a cocyclec(f ) ∈ Zn((K,L);πn(Y,B)). The cohomologyclass[c(f )] = 0 equals to 0 iff there exists a homotopyφ : (|K|, |L|) × [0,1] → (Y,B)

such that

φ(x, t) = f (x) for (x, t) ∈ (∣∣K(n−2)∣∣ ∪ |L|) × [0,1] ∪ |K| × {0}

and

φ(∣∣K(n)

∣∣ × {1}) ⊂ B.

If f1 : (K1,L1) → (K,L) is a simplicial map thenf |f1|(|K(n−1)1 | ∪ |L1|) ⊂ B and

[c(f |f1|)] = Hn(f1)([c(f )]).By the Universal Coefficient Theorem the mapfi : (Xi,Ai) → (Xi−1,Ai−1) induces

a trivial homorphismHi+r (fi) :Hi+r ((Xi−1,Ai−1);G) → Hi+r ((Xi,Ai);G) for everyAbelian groupG.

Applying Lemma 2.1 to the sequence(X0,A0)id→(X0,A0)

id→·· · id→(X0,A0) we find acellular mapf0 � id : (X0,A0) → (X0,A0) such thatf0(X

(r)0 ∪ A0) ⊂ A0. Hence we may

assume thatf1(X(r) ∪ A1) ⊂ A0.
1


uction] (see

evel

For everyi = 1,2, . . . , n there exists a cellular homotopy equivalenceα : (|Ki |, |Li |) →(Xi,Ai), whereKi is a simplicial complex andLi is a subcomplex ofKi .

Let βi : (Xi,Ai) → (|Ki |, |Li |) be the cellular homotopy inverse ofαi .Let g1 = f0f1α1 : (|K1|, |L1|) → (X0,A0). Theng1β1 � f1 andg1(|K(r)

1 | ∪ |L1|) ⊂A0.

Assume that fori � k < n we have constructed mapsgi : (|Ki |, |Li |) → (X0,A0) suchthat

giβi � f1f2 · · ·fi and gi

(∣∣K(r+i)i

∣∣ ∪ |Li |) ⊂ A0. (1)

The mapgkβkfk+1αk+1 : (|Kk+1|, |Lk+1|) → (X0,A0) is homotopic tof1f2 · · ·fk+1

andgkβkfk+1αk+1(|K(k+r+1)k+1 | ∪ |Lk+1|) ⊂ A0.

Since the mapfk+1 induces a trivial homorphism fromHk+r+1((Xk,Ak);πk+r+1(X0,A0)) to Hk+r+1((Xk+1,Ak+1);πk+r+1(X0,A0)), we infer that the cocyclec(gkβkfk+1αk+1) represents the trivial element of the groupHk+r+1((|Kk+1|, |Lk+1|);πk+1(X0,A0)) and thereforegkβkfk+1 can be replaced up to homotopy by a mapgk+1

satisfying (1) fori = k + 1.The mapg = gn satisfies the required conditions.�The next lemma is proved in [18]. It is consequence of algebraic methods (obstr

to compression) developed by E.H. Spanier and J.H.C. Whitehead in [19] and [20[19, p. 92] or [20, p. 359]).

Lemma 2.3. Let r be a natural number. Suppose that(Xi,Ai) is a pair of finite CWcomplexes fori = 0,1, . . . , n and fi : (Xi,Ai) → (Xi−1,Ai−1) is a cellular map fori = 1,2, . . . , n satisfying the following conditions:

(1) dimX0 � n + r ,(2) X0 andA0 are imply connected,(3) πn+r−i+1

s (fi) :πn+r−i+1s (Xi−1,Ai−1) → πn+r−i+1

s (Xi,Ai) is a trivial homomor-phism fori = 1,2, . . . , n.

Then for sufficiently largem there exists a mapg : (Sm(Xn),Sm(An)) → (Sm(X0),Sm(A0))

such thatSm(f1f2 · · ·fn) � g relSm(An) andg(Sm(Xn)) ⊂ Sm(X(m+r)0 ) ∪ Sm(A0).

3. Level morphisms

Every stable shape morphismf :X → Y can be represented (see [12, p. 12]) as a lmorphism belonging to Pro-SCWf (X,Y ), whereX = {Xσ pτ

σ } andY = {Yσ ,qτσ } are in-

verse systems ofSCWf over the same directed setΣ .Suppose thatf = {fσ }σ∈Σ ∈ Pro-SCWf (X,Y ) is a level morphism. For everyσ ∈ Σ

and everyτ > σ we have the diagram


mor-

l shapetion

se-

e

Xτfr

pτσ

Yτ

qτσ

Xσfσ

Yσ

in the categorySCWf .Let Mσ andMτ denote the reduced mapping cylinders of maps representing the

phismsfσ :Xσ → Yσ andfτ :Xτ → Yτ .Then there exists a map of spectrarτ

σ :Mτ → Mσ such that its restrictions toXτ andYτ equals (respectively) topτ

σ andqτσ .

We shall keep all these notations in the sequel.Proceeding analogously as in the proof of the Whitehead theorem for the classica

theory (see [12, p. 143], [16] or [6]) we can prove the following lemma. For complewe present the sketch of the proof.

Lemma 3.1. Suppose thatf ∈ Pro-SCWf (X,Y ) is a level morphism such that:

(1) Pro-πsn( f ) is a monomorphism.

(2) Pro-πsn+1( f ) is an epimorphism.

Then for everyσ ∈ Σ there existsτ > σ such thatPro-πsn+1(r

τσ ) = 0.

Proof. By induction (compare [12, p. 148, Lemma 4]) we can construct an infinitequenceσ = σ(0) < σ(1) < σ(2) < · · · of elements ofΣ such that Kerπs

n(fσ(i+1)) ⊂Kerπs

n(pσ(i+1)σ (i) ) and Imπs

n+1(qσ(i+1)σ (i) ) ⊂ Imπs

n+1(fσ(i)) for i = 0,1,2, . . . .

Then f ′ = {fσ i} : X′ = {Xσ(i),pσ(i+1)σ (i) } → Y′ = {Yσ(i),qσ(i+1)

σ (i) } is a level morphismsuch that Pro-πs

n(f ′) is a monomorphism and Pro-πsn+1(f

′) is an epimorphism.We have an exact sequence

· · · → πsn+1(Xσ(i)) → πs

n+1(Mσ(i)) → πsn+1(Mσ(i)), (Xσ(i))

→ πsn(Xσ(i)) → πs

n(Mσ(i)) → ·· ·for i = 0,1,2, . . . . These sequences together with the homorphisms induced bepσ(i+1)

σ (i)

andqσ(i+1)σ (i) form a tower of exact sequences

· · · → Pro-πsn+1(X ) → Pro-πs

n+1(M ) → Pro-πsn+1(M,X )

→ Pro-πsn(X ) → Pro-πs

n+1(M ) → ·· · ,i.e., an exact sequence of the category of progroups, where byM we denote the inverssequence{Mσ(i), rσ(i+1)

σ (i) }.We deduce (see [12, p. 118, Theorem 9]) that Pro-πs

n+1(M,X ) is a trivial progroup.This means that forσ = σ(0) there existsk such that Pro-πs

n+1(pτσ ) is a trivial homomor-

phism, whereτ = σ(k). �Lemma 3.2. Let f ∈ Pro-SCWf (X,Y ) be a level morphism andH = {hn} be a reducedgeneralized cohomology theory such that:


nd

that

(1) hn( f ) :hn(Y ) → hn(X ) is an epimorphism,(2) hn+1( f ) :hn+1(Y ) → hn+1(X ) is a monomorphism.

Then for eachσ ∈ Σ anda ∈ hn+1(Mσ ,Xσ ) there isτ > σ such thathn+1(rτσ )(a) = 0.

Proof. Consider the diagram

hn(Mσ )i1

jτ1

hn(Xσ )i2

jτ2

hn+1(Mσ ,Xσ )i3

jτ3

hn+1(Mσ )i4

jτ4

hn+1(Xσ )

jτ5

hn(Mτ )iτ1

hn(Xτ )iτ2

hn+1(Mτ ,Xτ )iτ3

hn+1(Mτ )iτ4

hn+1(Xτ )

(2)

where rows are long exact sequences (see, for example, [7, p. 7]) of(Mτ ,Xτ ) andH = {hn}.

Let a ∈ hn+1(Mσ ,Xσ ). We must show that there existsτ > σ such thatjτ3 (a) = 0.

If there isa′ ∈ hn(Xσ ) such thati2(a′) = a andjτ2 (a′) = 0 for someτ > σ , then we

haveiτ2(j τ2 (a′)) = jτ

3 (a) = 0.If jτ

2 (a′) �= 0 for everyτ > σ , then for someτ > σ there existsa′′ ∈ hn(Mτ ) such thatiτ1(a′′) = jτ

1 (a′).The exactness of the first and the second rows implies thatiτ2 iτ1(a′′) = 0 andjτ

3 (a) =iτ2 iτ1(a′′) = 0.

By the exactness of the first row ifi3(a) �= 0, then i4i3(a) = 0. Sincehn( f ) is amonomorphism, we get that there existsµ such thatjµ

4 (i3(a)) = 0. Theniµ2 (a′) = j

µ3 (a)

for somea′ ∈ hn(Xµ) and there existsτ > µ such thatjτ3 (a) = 0. �

Lemma 3.3. Let f ∈ Pro-SCWf (X,Y ) be a level morphism andH = {hn} be a re-duced generalized cohomology theory. If for eachσ ∈ Σ there is τ > σ such thathn+1(rτ

σ ) :hn+1(Mσ ,Xσ ) → (Mτ ,Xτ ) is a trivial homorphism, thenhn( f ) :hn(Y ) →hn(X ) is an epimorphism andhn+1( f ) :hn+1(Y) → hn+1(X) is a monomorphism.

Proof. Let us observe thatiτ2jτ2 = 0 for everya ∈ hn(Xσ ). By the exactness of the seco

row of the diagram (2) we get that there existsb ∈ hn(Mτ ) such thatiτ1(b) = jτ2 (a).

Suppose that[a] ∈ hn+1(Y ) is represented bya ∈ hn+1(Mσ ) ∼= hn+1(Yσ ) andhn+1( f )([a]) = 0. Then there existsτ > σ such thathn+1(rτ

σ ) is a trivial homorphismandiτ4jτ

4 (a) = jτ5 i4(a) = 0. The exactness of the second row of the diagram (2) gives

there existsb ∈ hn+1(Mτ ,Xτ ) such thatiτ3 = jτ4 (a). We conclude that[a] = 0. �

4. Deformability

Suppose thatf : (E,E0) → (F,F0) is a map of spectra. We say thatf is n-deformableinto F0 iff (compare [9, p. 203]) there exists a mapg � f : (E,E0) → (F,F0) such thatg(E(n) ∪ E) ⊂ F0.

A level morphismf ∈ Pro-SCWf (X,Y ) is calledn-deformable if for eachσ ∈ Σ thereis τ > σ such thatrτ

σ : (Mτ ,Xτ ) → (Mσ ,Xσ ) is n-deformable intoXσ . It is convenient to


or

l

a

t

extend this definition onto the case ofn = −∞ and to regard that everyf is n-deformablefor n = −∞.

Proposition 4.1. Let f ∈ Pro-SCWf (X,Y ) be ann-deformable level morphism. Then feveryσ ∈ Σ there isτ > σ such that the homomorphismHm(rτ

σ ) :Hm((Mσ ,Xσ );Z) →Hm((Mτ ,Xτ );Z) is trivia] for everym � n.

Proof. The maprτσ factors (up to homotopy) through(Mτ ,M

(n)τ ∪ Xτ ). �

Lemma 4.2. Letp < q be integers and letf ∈ Pro-SCWf (X,Y ) be ap-deformable levemorphism such that:

(1) Pro-πsp( f ) : Pro-πs

p(X ) → Pro-πsp(Y ) is a monomorphism.

(2) Pro-πsn( f ) : Pro-πs

n(X ) → Pro-πsn(Y ) is an isomorphism forp < n < q.

(3) Pro-πsq( f ) : Pro-πs

q(X ) → Pro-πsq(Y ) is an epimorphism.

Thenf is q-deformable.

Proof. Using Lemma 3.1 we can find a natural numberm(σ) and a finite sequenceσ =σ(0) < σ(1) < · · · < σ(q − p) such thatrσ(i+1)

σ (i) : (Mσ(i+1),Xσ(i+1)) → (Mσ(i),Xσ(i)) in-duces a trivial homorphism of the relative homotopy groups

πq+m−i

(rσ(i+1)σ (i)

):πq+m−i (Mσ(i+1),Xσ(i+1)) → πq+m−i (Mσ(i),Xσ(i)),

whereMσ(i+1),Xσ(i+1), Mσ(i) andXσ(i) are (respectively) themth terms of the spectr

Mσ(i+1),Xσ(i+1), Mσ(i) andXσ(i) andrσ(i+1)σ (i) is induced byrσ(i+1)

σ (i) .Increasingm if necessary (in this caseMσ(i) andXσ(i) becamer-connected, wherer

equals to the growth ofm) by the hypothesis we get that there existsµ > σ(q − p) = τ

and a mapg � rµτ : (Mµ,Xµ) → (Mτ ,Xτ ) such thatg(M

(m+p)µ ∪ Xµ) ⊂ Xτ .

Thenπi(Mµ,M(m+p)µ ∪ Xµ) = 0 for i = m + p,m + p − 1, . . . ,1.

Let us apply Lemma 2.1 for the case whenfi = rσ(i)σ (i−1) for i = 1, . . . , q − p, fq−p+1 =

g andfi = id : (Mµ,M(m+p)µ ∪Xµ) → (Mµ, M

(m+p)µ ∪Xµ) for i = q −p + 2, . . . , q +m.

As the result we obtain that there is a maph � pµσ such thath(M

(q+m)µ ∪Xµ) ⊂ Xσ . �

Lemma 4.3. Let q be an integer andp < q be an integer orp = −∞. Suppose also thaf ∈ Pro-SCWf (X,Y ) is ap-deformable level morphism such that:

(1) Hp( f ) :Hp(Y;Z) → Hp(X;Z) is an epimorphism.(2) Hn( f ) :Hn(Y;Z) → Hn(X;Z) is an isomorphism forp < n � q.(3) Hq+1( f ) :Hq+1(Y;Z) → Hq+1(X;Z) is a monomorphism.

Thenf is q-deformable.

Proof. We must show that for eachσ ∈ Σ and there isτ > σ such thatrτσ : (Mτ ,Xτ ) →

(Mσ ,Xσ ) is q-deformable intoXσ .


first of

en

The groupsHn((Mσ ,Xσ );Z) are finitely generated for everyn.Hence Lemma 3.2 guarantees that for everyτ ∈ Σ and everyn � q +1 there existsµ >

τ such that the induced homomorphismHn(pµτ ) :Hn((Mτ ,Xτ );Z) → Hn((Mµ,Xµ);Z)

is trivial.For σ ∈ Σ there existr and a pair ofr-connected CW complexes(Mσ ,Xσ ) suchMσ

andXσ that are themth terms ofMσ andXσ and forn > m thenth terms ofMσ andXσ

are(n − m)-fold suspensions ofMσ andXσ .Replacingm by a bigger one if necessary we can find a sequenceσ = σ(0) < · · · <

σ(q + m − r) such thatpσ(i)σ (i−1) : (Mσ(i),Xσ(i)) → (Mσ(i−1),Xσ(i−1)) induces trivial ho-

momorphisms

Hr+i(p

σ(i)σ (i−1)

):Hr+i

((Mσ(i−1),Xσ(i−1));Z

) → Hr+i((Mσ(i),Xσ(i));Z

)and

Hr+i+1(pσ(i)σ (i−1)

):Hr+i+1((Mσ(i−1),Xσ(i−1));Z

)

→ Hr+i+1((Mσ(i),Xσ(i));Z)

for i = 1, . . . , q + m − r.

Applying Lemma 2.2 for the case whenfi = pσ(i)σ (i−1) for i = 1, . . . , q+m−r we get that

there is a mapg � pτσ(0) such thatg(M

(m+q)τ ∪ Xτ ) ⊂ Xσ , whereτ = σ(q + m − r). �

Remark 1. Let us notice that the theses of Lemmas 4.2 and 4.3 are the same. In thethem we cannot assume thatp = −∞.

5. Compressibility

Suppose thatf : (E,E0) → (F,F0) is a map of spectra. We say thatf is n-compressibleinto F0 (compare [19]) iff there exists a mapg � f : (E,E0) → (F,F0) such thatg(E) ⊂E

(n) ∪ F0.A level morphismf ∈ Pro-SCWf (X,Y ) is calledn-compressible if for eachσ ∈ Σ

there isτ > σ such thatrτσ : (Mτ ,Xτ ) → (Mσ ,Xσ ) is n-compressible intoXσ .

We say thatf is q-compressible forq = −∞ iff for every σ there existτ > σ andg � rτ

σ : (Mτ ,Xτ ) → (Mσ ,Xσ ) such thatg(Mτ ) ⊂ Xσ .

Proposition 5.1. Let f ∈ Pro-SCWf (X,Y ) be ann-compressible level morphism. Thfor everyσ ∈ Σ there isτ > σ such that the homomorphismHm(rτ

σ ):

Hm(Mσ ,Xσ ;Z) → Hm(Mτ ,Xτ ;Z) is trivial for everym > n.

Proof. The maprτσ factors (up to homotopy) through(M(n)

σ ∪ Xσ ,Xσ ). �Lemma 5.2. Let q be an integer orq = −∞ and let f ∈ Pro-SCWf (X,Y ) be a levelmorphism such that:

(1) πqs ( f ) :πq

s (Y ) → πqs (X ) is an epimorphism.

(2) πns ( f ) :πn

s (Y ) → πns (X ) is an isomorphism forn > q.

Thenf is q compressible.


1.1]

at

Proof. For σ ∈ Σ there exists a pair of simply connected CW complexes(Mσ ,Xσ ) suchMσ andXσ are themth terms ofMσ andXσ and forn > m thenth terms ofMσ andXσ

are the(n − m)-fold suspensions ofMσ andXσ .Let M0 = dimMσ − m − q. There exists a sequenceσ = σ(0) < · · · < σ(m0) such that

the induced homomorphisms

πm+q+1+m0−is

(rσ(i)σ (i−1)

):πm+q+1+m0−i

s (Mσ(i−1)/Xσ(i−1))

→ πm+q+1+m0−is (Mσ(i)/Xσ(i))

are trivial fori = 1,2, . . . ,m0.Applying Lemma 2.3 for the case whenr = m + q, n = m0 andfi = r

σ(i)σ (i−1) for i =

1,2, . . . ,m0 we get that for sufficiently largel andτ = σ(m0) there exists a map

g :(Sl (Mτ ),Sl (Xτ )

) → (Sl(Mσ ),Sl (Xσ )

)

such thatg � Sl(rτσ ) relSl(Xτ ) andg(Sl (Mτ )) ⊂ Sl(M

(m+q)σ ) ∪ Sl(Xσ ).

This means thatf is q-compressible. �

6. Whitehead theorem for Pro -SCWf

Theorem 6.1. Suppose thatq is an integer orq = −∞. Everyq-deformable andq-com-pressible level morphism is an isomorphism.

Proof. In order to show thatf is an isomorphism it suffices to show (see [16, Theoremor [6, p. 7, Theorem 2.3]) that for everyσ there existτ > σ and g � rτ

σ : (Mτ ,Xτ ) →(Mσ ,Xσ ) such thatg(Mτ ) ⊂ Xσ , i.e.,f is q-compressible forq = −∞.

Let q be an integer. The morphismf ∈ Pro-SCWf (X,Y ) is aq-deformable andq-com-pressible. Then for sufficiently largem and for eachσ ∈ Σ there exist a pair(τ,µ) withσ < t < µ and mapsg � pτ

σ : (Mτ ,Xτ ) → (Mσ ,Xσ ), h � pµτ : (Mµ,Xµ) → (Mτ ,Xτ )

such thatg(M(q+m)τ ∪ Xτ ) ⊂ Xσ andh(Mµ) ⊂ M

(q+m)τ ∪ Xτ .

The mapgh � pµσ : (Mµ,Xµ) → (Mσ ,Xσ ) andgh(Mµ) ⊂ Xσ . �

Theorem 6.1 allows to produce algebraic criteria for checking thatf ∈ Pro-SCWf (X,Y )

is an isomorphism.

Theorem 6.2. Let p be an integer orp = −∞ and q � p be an integer. Suppose thf ∈ Pro-SCWf (X,Y ) is a morphism such that:

(1) Hn( f ) :Hn(Y;Z) → Hn(X;Z) is an isomorphism forn � p.(2) Hp+1( f ) :Hp+1(Y;Z) → Hp+1(X;Z) is a monomorphism.(3) Pro-πs

p( f ) : Pro-πsp(X ) → Pro-πs

p(Y ) is a monomorphism.(4) Pro-πs

n( f ) : Pro-πsn(X ) → Pro-πs

n(Y ) is an isomorphism forp � n < q.(5) Pro-πs

q( f ) : Pro-πsq(X ) → Pro-πs

q(Y ) is an epimorphism.

(6) πqs ( f ) :πq

s (Y ) → πqs (X ) is an epimorphism.

(7) πns ( f ) :πn

s (Y ) → πns (X ) is an isomorphism forn > q.


d

.

Thenf is an isomorphism. Ifq � p + 1 everyf satisfying(1), (2), (6)and (7) is anisomorphism.

Proof. Without restriction of generality we may assume (see [12, p. 12]) thatf is a levelmorphism.

By Lemmas 4.3 and 5.2 the morphismf is p-deformable andq-compressible anLemma 4.2 guarantees that it isq-deformable.

Let assume thatq = p + 1.By Proposition 5.1 we get that for everyσ there existsτ > σ such that

Hq+1(rτσ ) :Hq+1(Mσ ,Xσ ;Z) → Hq+1(Mτ ,Xτ ;Z)

is a trivial homorphism.Lemma 3.3 gives thatHq( f ) is an epimorphism andHq+1( f ) is a monomorphism

This means thatHq( f ) is an isomorphism and from Lemma 4.3 we conclude thatf isq-deformable. �

7. Whitehead theorem for ShStab

Every morphism of the category ShStab(X,Y ) may be identified with the morphismf ∈Pro-SCWf (X,Y ). We may also assume thatHm(Xσ ;Z) = 0= Hm(Yσ ;Z) for m < 0.

Theorem 7.1. The morphismf ∈ ShStab(X,Y ) is an isomorphism iff there existsq suchthat the following conditions are satisfied:

(a) πqs ( f ) :πq

s (Y ) → πqs (X) is an epimorphism.

(b) πns ( f ) :πn

s (Y ) → πns (X) is an isomorphism forn > q.

(c) Hn( f ) :Hn(Y ;Z) → Hn(X;Z) is an isomorphism for everyn < q.(d) Hq( f ) :Hq(Y ;Z) → Hq(X;Z) is a monomorphism.

Proof. Theorem is a consequence of Theorem 6.2.�Theorem 7.2. The morphismf ∈ ShStab(X,Y ) is an isomorphism iff there existsq suchthat the following conditions are satisfied:

(a) πns ( f ) :πn

s (Y ) → πns (X) is an isomorphism for everyn > q.

(b) πsq( f ) :πs

q(Y ) → πsq(X) is an epimorphism.

(c) Pro-πsq( f ) : Pro-πs

q(X) → Pro-πsq(Y ) is an isomorphism.

(d) Pro-πsn( f ) : Pro-πs

n(X) → Pro-πsn(Y ) is an isomorphism for everyn < q.

Proof. The morphism of the category ShStab(X,Y ) may be identified with

f ∈ Pro-SCWf (X,Y )

such thatHm(Xσ ;Z) = 0= Hm(Yσ ;Z) for m < 0.Hence the conditions (1) an (2) from Theorem 6.2 are automatically satisfied.�


egory

y

hape

l

rallyn

t

f

8. Final remarks

Most of the theorems of the stable homotopy category hold for the homotopy catof simply connected CW complexes, however, their proofs often are much longer.

In particular, a mapf :P → Q is a homotopy equivalence ifff is a stable homotopequivalence.

Analogously, ifX andY are shape 1-connected finite-dimensional continua then a smorphismf :X → Y is a shape equivalence ifff induces isomorphisms of allCech coho-mology groups with coefficients inZ. This means that ifX andY are finite-dimensionathenf is a shape equivalence if and only iff is a stable shape equivalence.

Problem 1. Suppose thatX and Y are shape 1-connected continua (or more geneHausdorff continua). Is it true that the shape morphismf :X → Y is an isomorphism whef is a stable shape equivalence?

Problem 2. Suppose thatX is a shape 1-connected continuum with max{n: πns (X) �= 0} <

∞. Is it true that Sh(X) = Sh(Y ), where dimY < ∞?

The cohomotopical dimensionπ- dimX of X is less then or equal ton iff (see [4,5])the inclusioni :A → X induces an epimorphismπn

s (i) :πns (X) → πn

s (A) for every closedsubsetA of X.

We know (see [18]) thatπ- dimX � dimZ X and thatπ- dim = ∞ if π- dimX �=dimZ X.

Problem 3. Is it true that the cohomotopical dimensionπ- dimX = dimX for everyX?

Theorem 8.1. If there exists a compactumX with dimX = ∞ and π-dimX < ∞, thenthere exists a continuumY such that:

(i) Y is shapek-connected andπks (Y ) = 0 for everyk = 1,2, . . . .

(ii) The shape dimensionFd(Y ) = ∞ (i.e., Sh(Y ) �= Sh(A) for every compactumA withdimA < ∞).

Proof. Let π - dimX = m. Since dimX = ∞, we infer that there exists a closed subseA

of X and a mapf :A → Sm+10 which is not extendable overX. We know thatf is anessential map,π - dimA � m andπk

s (A) = 0 for k > m.Thenπ- dimA = dimZ A � m and there exists a CE maph :P → A of compactumP

with dimP � m ontoA.Let Y be the mapping cone ofh (i.e., the space obtained from the mapping cylinder oh

by collapsingP × {0} to a point).There exists a CE mapg :D → Y of the coneD overP ontoY such thatg(x) = h(x)

for x ∈ P andg(x) = x for x ∈ D\P .The mapg :D → Y induces isomorphisms of cohomology groupsHk(g) :Hk(Y ;Z) →

Hk(D;Z) and homology progroups Pro-πk(g) : Pro-πk(D) → Pro-πk(Y ) for all basepoints ofD and allk.


-

nd

974.

ometric

7–262.ppl. 47

) 1–51.

9) 91–

5) 139–

(1974)

f

100.y60.

Hence the continuumY is shapek-connected and

Hk(Y ;Z) = 0 for everyk > 0. (3)

The spaceY is homeomorphic to the union of the mapping cylinderM ⊃ A ∪ P ofh :P → A and the coneC overP , C ∩ M = P .

The Mayer–Vietoris sequences for the stable cohomotopy theory implies that

πks (Y ) = 0 for k > m. (4)

The mapf :A → Sm+10 is extendable overM . Let f :M → Sm+10 denotes the extension off .

Next from dimP � m we conclude that there exists a mapf0 :Y → Sm+10 such thatf0(x) = f (x) for x ∈ M .

We have

f0 :Y → Sm+10 is an essential map. (5)

From (3) and (5) we conclude that (see [12, p. 251]) that Fd(Y ) = ∞.By Theorem 7.1, (3) and (4) we see thatg :D → Y is a stable shape equivalence a

max{n: πns (Y ) �= 0} = 0. �

References

[1] J.F. Adams, Stable Homotopy and Generalized Homology, University of Chicago Press, Chicago, 1[2] K. Borsuk, Theory of Shape, Polish Scientific Publishers, Warsaw, 1975.[3] A. Dold, D. Puppe, Duality, trace and transfer, in: Proceedings of the International Conference on Ge

Topology, Warszawa, 1980, pp. 81–102.[4] A.N. Dranisnikov, Generalized dimension of compact metric spaces, Tsukuba J. Math. 14 (1990) 24[5] A.N. Dranisnikov, Stable cohomotopy dimension and weakly infinite dimensional spaces, Topology A

(1992) 79–81.[6] J. Dydak, The Whitehead and the Smale theorems in shape theory, Dissertationes Math. 156 (1979[7] E. Dyer, Cohomology Theories, Benjamin, New York, 1969.[8] H.W. Henn, Duality in stable shape theory, Arch. Math. 36 (1981) 327–341.[9] S.T. Hu, Homotopy Theory, Academic Press, San Diego, CA, 1959.

[10] S.T. Hu, Theory of Retracts, Wayne State University Press, Detroit, 1965.[11] E.L. Lima, The Spanier–Whitehead duality in new homotopy categories, Summa Bras. Math. 4 (195

148.[12] S. Mardešic, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.[13] H.R. Margolis, Spectra and the Steenrod Algebra, North-Holland, Amsterdam, 1983.[14] T. Miyata, Generalized stable shape and duality, Topology Appl. 109 (2001) 75–88.[15] T. Miyata, J. Segal, Generalized stable shape and the Whitehead theorem, Topology Appl. 63 (199

164.[16] K. Morita, The Hurewicz and the Whitehead theorems in shape theory, Sci. Rep. Kyoiku Daigaku 12

246–258.[17] S. Nowak, On the relationships between shape properties of subcompacta ofSn and homotopy properties o

their complements, Fund. Math. 128 (1987) 47–60.[18] J.S. Nowak, Stable cohomotopy groups of compact spaces, Fund. Math. 180 (2003) 99–137.[19] E.H. Spanier, J.H.C. Whitehead, Obstructions to compression, Quart. J. Math. Oxford 6 (1955) 91–[20] E.H. Spanier, J.H.C. Whitehead, The theory of carriers andS-theory, in: Algebraic Geometry and Topolog

(A Symposium in honor of S. Lefschetz), Princeton University Press, Princeton, NJ, 1956, pp. 330–3[21] R.M. Switzer, Algebraic Topology—Homotopy and Homology, Springer, Berlin, 1975.

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