Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups
In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the...
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irk-123456789-1656402020-02-16T01:26:23Z Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups Golasinski, M. Goncalves, D. Wong, P. Статті In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [∑ (V×WU∗), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. Уточнено означення торових гомотопічних груп Фокса, доведено розщеплення точної послідовності цих груп. Наведено означення оціночних підгруп і знайдено їх зв'язок із класичними підгрупами Готтліба. На основі цих конструкцій встановлено деякі властивості груп Абе та доведено деякі результати Готтліба для оціночних підгруп гомотопічних груп Фокса. Наведено подальше узагальнення груп Фокса та означення групи τ = [∑ (V×WU∗), X], у якій узагальнення Арковича добутку Уайтхеда також є комутатором. Насамкінець показано, що узагальнена група Готтліба міститься у центрі групи τ, що покращує результат Варадараяна. 2005 Article Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups / M. Golasinski, D. Goncalves, P. Wong // Український математичний журнал. — 2005. — Т. 57, № 3. — С. 320–328. — Бібліогр.: 16 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165640 515.143 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Golasinski, M. Goncalves, D. Wong, P. Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups Український математичний журнал |
| description |
In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [∑ (V×WU∗), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. |
| format |
Article |
| author |
Golasinski, M. Goncalves, D. Wong, P. |
| author_facet |
Golasinski, M. Goncalves, D. Wong, P. |
| author_sort |
Golasinski, M. |
| title |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups |
| title_short |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups |
| title_full |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups |
| title_fullStr |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups |
| title_full_unstemmed |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups |
| title_sort |
generalizations of fox homotopy groups, whitehead products, and gottlieb groups |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165640 |
| citation_txt |
Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups / M. Golasinski, D. Goncalves, P. Wong // Український математичний журнал. — 2005. — Т. 57, № 3. — С. 320–328. — Бібліогр.: 16 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT golasinskim generalizationsoffoxhomotopygroupswhiteheadproductsandgottliebgroups AT goncalvesd generalizationsoffoxhomotopygroupswhiteheadproductsandgottliebgroups AT wongp generalizationsoffoxhomotopygroupswhiteheadproductsandgottliebgroups |
| first_indexed |
2025-07-14T19:17:56Z |
| last_indexed |
2025-07-14T19:17:56Z |
| _version_ |
1837651122616008704 |
| fulltext |
UDC 515.143
M. Golasinski′ (Nicholas Copernicus Univ., Torun′ , Poland),
D. Gonçalves (IME-USP, Brasil),
P. Wong (Bates College, Lewiston, USA)
GENERALIZATIONS OF FOX HOMOTOPY GROUPS,
WHITEHEAD PRODUCTS AND GOTTLIEB GROUPS*
UZAHAL|NENNQ HOMOTOPIÇNYX HRUP FOKSA,
DOBUTKY UAJTXEDA I HRUPY HOTTLIBA
In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence
of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups.
With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation
subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [ Σ ( V ×
× W ∪ * ), X ] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally,
we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of
Varadarajan.
Utoçneno oznaçennq torovyx homotopiçnyx hrup Foksa, dovedeno rozweplennq toçno] poslidov-
nosti cyx hrup. Navedeno oznaçennq ocinoçnyx pidhrup i znajdeno ]x zv’qzok iz klasyçnymy
pidhrupamy Hottliba. Na osnovi cyx konstrukcij vstanovleno deqki vlastyvosti hrup Abe ta
dovedeno deqki rezul\taty Hottliba dlq ocinoçnyx pidhrup homotopiçnyx hrup Foksa. Navedeno
podal\ße uzahal\nennq hrup Foksa ta oznaçennq hrupy τ = [ Σ ( V × W ∪ * ), X ] , u qkij uza-
hal\nennq Arkovyça dobutku Uajtxeda takoΩ [ komutatorom. Nasamkinec\ pokazano, wo
uzahal\nena hrupa Hottliba mistyt\sq u-centri hrupy τ, wo pokrawu[ rezul\tat Varadaraqna.
Introduction. In 1941, J. H. C. Whitehead [1] introduced the notion of a product,
between elements of the higher homotopy groups of a space, now known as the
Whitehead product. It is well known that if α, β ∈ π1( )X then the Whitehead product
α � β of α and β is simply the commutator [α, β] . In an attempt to give a more
geometric description of the Whitehead product, R. Fox introduced in [2] the torus
homotopy groups. Given a path connected space X, the n-th torus homotopy group
τn X( ) is isomorphic to the fundamental group of the function space XT n−1
, where
T n−1 denotes the (n – 1)-dimensional torus. In [2], an elegant and geometric
description of the elements of τn X( ) was given. Take for example when n = 2, a
typical element of τ2 is the homotopy class of maps of the form f : F2 → X where
F2 is the pinched 2-torus, i.e., F2 is the quotient of S S1 1× by S1 × { }s0 for some
basepoint s S0
1∈ . It follows that F2 has the same homotopy type as the reduced
suspension of S1 ∪ *( ) , the disjoint union of the circle with a distinguished point.
With this description of the Fn as the pinched n-torus, we are led to extend the
definition of τn. We reprove the main results of [2] using modern language of
homotopy theory. Our approach allows us to generalize many results concerning
Gottlieb groups or generalized evaluation subgroups. The insight from [2] sheds new
light into the generalized Whitehead product given by M. Arkowitz [3]. In particular,
we show, in the same spirit as in [2], another way so that the generalized Whitehead
product when embedded in a larger (and different) group is a commutator as well.
Although R. Fox introduced his so-called torus homotopy groups [2] (first
announced in 1945) in 1948 and made a connection with the Whitehead products, these
* This work was conducted at the International Stefan Banach Center in Warsaw during the period
September 20 – 30, 2003. The authors wish to thank the Banach Center for its hospitality. The second
author’s travel was partially supported by the „Projeto temático Topologia Algébrica e Geométrica-
FAPESP, no 00/05385.8” and the third author’s travel was partially supported under a Philips Fellowship
from Bates College.
© M. GOLASINSKI′ , D. GONÇALVES, P. WONG, 2005
320 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
GENERALIZATIONS OF FOX HOMOTOPY GROUPS, WHITEHEAD PRODUCTS … 321
homotopy groups seem to have been forgotten in the development of algebraic
topology. It is the purpose of this paper to show that Fox’s concept of the torus
homotopy groups can be used to generalize and to unify many other results such as
those in [3], and in [4].
This paper is organized as follows. In Section 1, we redefine the torus homotopy
groups of Fox and reprove the main results of [2]. In Section 2, we study the
evaluation subgroups of the Fox groups, following the work of D. Gottlieb [5]. In
Section 3, we extend the definition of the Fox groups and obtain a similar split exact
sequence (Theorem 3.1). In Section 4, we relate our generalized Fox groups of
Section 3 with the generalized Whitehead product of Arkowitz [3]. In particular, we
introduce a group τ in which Arkowitz’s product is again a commutator
(Theorem 4.1). Moreover, we show that the generalized Gottlieb group lies in the
center of τ (Theorem 4.2).
Throughout, all spaces are assumed to be compactly generated as in [6]. This
assumption is made solely for the fact that the two definitions of the Gottlieb groups,
namely, the one defined using associated maps, and the one using the evaluation map,
are indeed isomorphic in the category of compactly generated spaces.
1. Fox torus homotopy groups. Using modern language of homotopy theory, we
redefine in this section Fox’s torus homotopy groups and we improve upon Fox’s
results.
Definition 1.1. Let X be a space and x X0 ∈ a basepoint. For n ≥ 1, the n-
th Fox group of X is defined to be
τn
nX x T X( , ) * ,0
1= ( )[ ]−Σ ∪ ,
where T k denotes the k-dimensional torus, Σ denotes the reduced suspension,
and [ , ] denotes the set of homotopy classes of based point preserving maps.
When defining homotopy groups, one considers basepoint homotopy classes of
maps from spheres to a space. In a similar fashion, one can interpret the Fox torus
homotopy groups as basepoint homotopy classes of maps from the suspension of tori
with an extra basepoint to a given space. Thus, we call an (n-dimensional) pinched
torus Σ T n−( )1 ∪ * a Fox space, denoted by Fn.
Proposition 1.1. The suspension of a Fox space has the homotopy type of a
bouquet of spheres. More precisely,
Σ T T Sk k
k k
− −
=
−
−
( ) ≈ ( )1 2
2
2
2/ V
�
� � .
Proof. Let us consider the following Barratt – Puppe sequence (see e.g. [7] or [8]
T k−2 → T k−1 → T Tk k− −1 2/ → Σ T k−2 → Σ T k−1 → Σ T Tk k− −( )1 2/ → …
associated with the cofibration T k−2 ⊂→ T k−1. Using the formula
Σ Σ Σ ΣX Y X Y X Y×( ) ≈ ∨ ∨ ∧( ) (1.1)
we obtain that Σ T Sk− ×( )2 1 has the same homotopy type as
Σ Σ ΣT S T Sk k− −∨ ∨ ∧( )2 1 2 1
where Σ T Sk− ∧( )2 1 ≈ Σ2 2T k−( ). Since the projection X × Y → X is a left inverse of
the inclusion X → X × Y, it follows that Σ T Tk k− −( )1 2/ has the same homotopy type
as Σ ΣS T k1 2 2∨ ( )− . On the other hand, the suspension of the torus T m = S
m1( ) , by
using the formula (1.1) for the suspension of a product, has the homotopy type of a
wedge of spheres where the number of spheres in dimension � is given by the
binomial coefficient
m
� −( )2
. By taking the suspension again, it is straightforward to
deduce the desired formula.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
322 M. GOLASINSKI′ , D. GONÇALVES, P. WONG
Theorem 1.1. Let X be a path connected space. Then
0 11 1→ ( ) → → →− −τ τ τn n nX X XΩ ( ) ( )
–�
(1.2)
is split exact. Moreover,
τ π α
n
i
n
iX X i
−
=
( ) ≅ ∏1
2
Ω ( ) ,
where αi is the binomial coefficient
n
i
−
−
2
2 .
Proof. Consider the Barratt – Puppe sequence
T
n−2 ∪* → T
n−1 ∪* → T Tn n− −1 2/ →
→ Σ T n−( )2 ∪* → Σ T n−( )1 ∪* → Σ T Tn n− −( )1 2/ → …
associated with T k−2 ∪ * ⊂→ T k−1 ∪ * and the long exact sequence
… → Σ T T Xn n− −( )[ ]1 2/ , → Σ T Xn−( )[ ]1 ∪* , → Σ T Xn−( )[ ]2 ∪* , → …
of groups by taking basepoint homotopy classes of maps into a space X . The
projection T n−1 → T n−2 onto the first (n – 2) coordinates induces a basepoint
preserving map T n−( )1 ∪* → T n−( )2 ∪* , and hence a homomorphism τn Y−1( ) →
→ τn Y( ) which is a right inverse of the homomorphism τn Y( ) → τn Y−1( ) for any
space Y. Consequently, the exactness and the splitting of the short exact sequence
(1.2) follow. The second part follows from Proposition l.1 in a straightforward manner.
Not only does Theorem 1.1 contain the following result of Fox, it also expresses the
kernel of the short exact sequence (1.2) in terms of torus homotopy groups. In [2], Fox
proved the following theorem.
Theorem 1.2. Let X be a space. Then
0 →
i
n
i X i
=
∏
2
π α( ) → τn X( ) →�–
τn X−1( ) → 1 (1.3)
is split exact where αi =
n
i
−
−
2
2
.
As indicated in [2], Theorem 1.2 asserts, in particular, that we have
τ1( )X ⊆ τ2( )X ⊆ τ3( )X ⊆ … , (1.4)
where the inclusions are the sections as in (1.3).
To conclude this section, we ask how the torus homotopy groups are related with
respect to fibrations. We obtain the following theorem.
Theorem 1.3. Let F ⊂→ E → B be a fibration. For any positive integer k ,
there is a long exact sequence
… → τk
nFΩ( ) → τk
nEΩ( ) → τk
nBΩ( ) →
d* τk
n FΩ −( )1 → … .
Proof. Consider the long sequence
… → ΩnF → ΩnE → ΩnB →
d* Ωn F−1 → …
associated to a fibration (see e.g. [8] F ⊂→ E → B. By taking homotopy classes of
maps from a space X into the spaces of this sequence, we obtain a long exact
sequence. When X is the Fox space Fk
, we obtain the desired sequence.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
GENERALIZATIONS OF FOX HOMOTOPY GROUPS, WHITEHEAD PRODUCTS … 323
2. Gottlieb – Fox groups. In [5], D. Gottlieb introduced the so-called (classical)
Gottlieb groups G Xn( ) of a space X as the set of maps Sn ∨ X → X that can be
extended to Sn × X → X. He showed that, when X has the homotopy type of a CW
complex, G Xn( ) is the image of the homomorphism induced by the evaluation map
e v : X X
X, 1( ) → X x, 0( ) on πn X x, 0( ). In the category of compactly generated
spaces, these two notions are equivalent.
Definition 2.1. Let X be a space and x X0 ∈ . For n ≥ 1, the n-th Gottli-
eb – Fox group of X is defined to be the evaluation subgroup, denoted by G Xnτ ( ),
i.e., the image of the evaluation homomorphism. Thus,
G Xnτ ( ) : =
Im : , ,e X X xn
X
X nv
*
τ τ1 0( ) → ( )( ) .
In general, τn X( ) is not abelian so it is not clear whether G Xnτ ( ) would be. As
it turns out, the Gottlieb – Fox groups are indeed abelian and can be expressed in terms
of the classical Gottlieb groups. Moreover, we will show that G Xnτ ( ) is central
in τn X( ) .
Theorem 2.1. The Gottlieb – Fox group is a direct product of ordinary Gottlieb
groups. In fact, we have
G Xnτ ( ) =
i
n
iG X i
=
∏
1
( )γ ,
where γ i is the binomial coefficient
n
i
−
−
1
1
.
Proof. First note that Ω X X( ) ≈ Ω X X( ). Consider the commutative diagram
τ τ τ
τ τ τ
τ τ τ
n
X e
n n
n
X e
n n
n
X e
n n
X G X X
X G X X
X G X X
− −
⊆
−
⊆
− −
⊆
−
( ) → ( ) → ( )
↓ ↓ ↓
( ) → ( ) → ( )
↓ ↓ ↓
( ) → ( ) → ( )
1 1 1
1 1 1
Ω Ω Ωv
v
v
*
*
*
proj. proj. proj.
,
where the vertical columns arc short split-exact sequences as in Theorem l.1. The fact
that G Xnτ − ( )1 Ω is the kernel of the middle vertical sequence follows from a diagram
chasing argument. Moreover, G Xnτ − ( )1 Ω =
i iG X i
=∏ 2
( )α so that
G Xnτ ( ) ≅
i
iG X i
=
∏
2
( )α �| G Xnτ − ( )1 .
By induction and the fact that the action becomes conjugation in τn X( ) , the semi-
direct product is in fact a direct product. Finally, an easy combinatorial argument
shows that
j
n k n j
k
=
− +
∑ −
−
2
2
2
= γ k
and the formula for G Xnτ ( ) follows.
In [5], it was proved that the Whitehead product of the elements of the Gottlieb
group with any other element in the higher homotopy groups is zero. This together
with the interpretation of the Whitehead product given by Fox in [2] yields the
following corollary.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
324 M. GOLASINSKI′ , D. GONÇALVES, P. WONG
Corollary 2.1. Denote by Z G( ) the center of a group G. Then,
G Xnτ ( ) ⊂ Z Xnτ ( )( ),
i.e., G Xnτ ( ) is central in τn X( ) .
In general, there is no relationship among the classical Gottlieb groups { }Gn . On
the other hand, the nested property of the torus homotopy groups (1.4) also holds for
the Gottlieb – Fox groups, i.e., G nτ −1 ⊆ G nτ .
In [5] the elements of the Gottlieb group were shown to contain the image of the
boundary homomorphism in the long exact sequence of homotopy groups of a
fibration. We will show a similar result for the torus homotopy groups.
Theorem 2.2. Let F ⊂→ E → B be a fibration, then
d B G Fn n* ( )τ τΩ( ) ⊂ ( ),
where d* is induced by the action of Ω B on F. Furthermore, the group G Fnτ ( )
is the union of the images of the homomorphism d* : τn B( ˜)Ω → τn F( ) where the
union is taken over all fibration F ⊂→ Ẽ → B̃ having the some fiber F.
Proof. The action of Ω B on F determines a map D : Ω B → FF up to
homotopy. The map d then factors through FF and d = e v ° D up to homotopy,
where e v is the evaluation map. For the second part, consider the universal fibration
over the base B̃ = B Aut (F ), the classifying space for the group Aut (F ). The image
of the boundary homomorphism is exactly the image of the evaluation given in
Definition 2.1 and the result follows.
3. Generalization of the functor ττn . One can define a generalization of the Fox
torus group as follows.
Definition 3.1. Let X be a space and x X0 ∈ a basepoint. For any space W,
the W-Fox group of X is defined to be
τW X x W X( , ) ( *),0 = [ ]Σ ∪ ,
where Σ denotes the reduced suspension, and [ , ] denotes the set of homotopy
classes of basepoint preserving maps.
First of all, we have the following useful description of τW X x( , )0 .
Lemma 3.1. For any X and W, we have
τ π πW
WX x X c W X X x( , ) , , | ( , )0 1 0 1 0≅ ( ) ≅ [ ]Σ � ,
where c0 : W → X is the constant map at x0 .
Proof. For any X and W, we have a fibration
X X XW W
* → → ,
where XW
* is the function space of basepoint preserving maps. This fibration admits a
section and the based point in XW is the constant map. We then obtain a long exact
sequence similar to that of Theorem l.3. The last three terms of the resulting long exact
sequence together with the splitting gives the result.
Remark 3.1. We should point out that several authors have studied the track
groups in W-topology (see e.g. [9] and subsequent works such as [10 – 12]). In these
works, the identification Σk W X,[ ] ≅ πk
WX( ) is used, where XW is understood to
be the set of based-point preserving maps from W to X whereas the notation XW
used in this paper denotes the set Map ( W, X ) of all maps from W to X . For
example, XS1
is the free loop space Λ X here but XS1
= Ω X in [9].
As an immediate consequence of Lemma 3.1, we show that the Abe groups of [13]
are semi-direct products. More precisely, the n-th Abe group κn X( ), which is defined
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
GENERALIZATIONS OF FOX HOMOTOPY GROUPS, WHITEHEAD PRODUCTS … 325
to be the fundamental group of the function space XSn−1
, is the split extension of
πn X( ) by π1( )X .
Corollary 3.1. The Abe groups are semi-direct products, i.e.,
κn X( ) = τ
Sn X−1 ( ) = πn X( ) �| π1( )X .
Remark 3.2. Note that there is a canonical map T n−1 ∪ * → Sn−1 by collapsing
the (n – 2)-th skeleton of T n−1 ∪ * . This induces a monomorphism τ
Sn X−1 ( ) →
τn X( ) . This fact that τn X( ) contains an isomorphic copy of κn X( ) was already
established in [2|.
Next, we give a generalization of Fox’s theorem (1.2) as follows.
Theorem 3.1. For any path connected X, V and W, the following sequence:
1 1→ ×( )[ ] → → →×V W V X X XV W V/ , ( ) ( )
–Ω τ τ�
(3.1)
is split exact. If W = Σ A is a suspension, then V W V X×( )[ ]/ , Ω is abelian and is
isomorphic to V W X∧[ ], Ω × W X, Ω[ ].
Proof. From the following Barratt – Puppe sequence
V ∪ * → V W×( ) ∪ * → V W V×( ) / → Σ V ∪ *( ) →
→ Σ V W×( )( )∪ * → Σ V W V×( )( )/ → …
and the fact that the map V → V W×( ) admits a left inverse, by taking homotopy
classes of maps into X, we obtain the following split exact sequence of groups:
1 1→ ×( )( )[ ] → ×( )( )[ ] → ( )[ ] →Σ Σ ΣV W V X V W X V X/ , * , * ,
–∪ ∪�
.
Now we study the group Σ V W V X×( )( )[ ]/ , .
Consider the Barratt – Puppe sequence
W V W V V W W V W V V W→ ×( ) → ∧ → → ×( )( ) → ∧( ) → …/ /Σ Σ Σ
and its associated sequence of groups by taking homotopy classes of maps into X.
Since the map W → V W V×( ) / admits a left inverse we obtain the following
sequence which is split exact:
1 1→ ∧[ ] → ×( )[ ] → [ ] →V W X V W V X W X, , ,/ –Ω Ω Ω�
. (3.2)
It follows that
Σ V W V X×( )( )[ ]/ , ≅ V W X∧[ ], Ω �| W X, Ω[ ].
Since
Σ Σ Σ ΣV W V W V W×( ) ≈ ∨ ∨ ∧( ),
it follows that
Σ ΣW V W V W V∨ ∧( )( ) ≈ ×( ) / .
Now, if W = Σ A for some A, then
Σ ΣW V W V W V∨ ∧( )( ) ≈ ×( ) /
is a double suspension and thus Σ W V W X∨ ∧( )( )[ ], is abelian with the group
structure given by the double suspension loop structure. There are canonical
projections
α: /V W V W×( ) → and β: /V W V V W×( ) → ∧ .
The co-multiplication on W = Σ A gives rise to a co-multiplication
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326 M. GOLASINSKI′ , D. GONÇALVES, P. WONG
ν: / / /V W V V W V V W V×( ) → ×( )( ) ∨ ×( )( ).
Together with α and β, we have a suspension map
Σ ΣV W V W V W×( ) → ∨ ∧( )( )/
which induces an isomorphism on homology and hence on homotopy classes since the
spaces are simply connected. Also the suspension map induces a group
homomorphism
Σ ΣW V W X V W V X∨ ∧( )( )[ ] → ×( )[ ], ,/ .
Therefore, we have an isomorphism and this implies that the semi-direct product (3.2)
is indeed a direct product.
Remark 3.3. If W = S1, V = T n−2 , then (3.1) becomes (1.2) since V W V×( ) =
= T Tn n− −1 2/ is the Fox space Fn−1 (pinched torus) so that V W V×( ) / ≈
≈ Σ T n−( )2 ∪ * .
4. Generalized Whitehead products and Gottlieb groups. In [3], a generalized
Whitehead product was defined between elements of ΣA X,[ ] and of ΣB X,[ ]. Here,
we give further information concerning Arkowitz’s product with the insight gained
from the Fox torus homotopy groups. We also consider the generalized Gottlieb group
with respect the functor τW .
We start by recalling the definition of the generalized Whitehead product given in
[3]. Let f : Σ A → X, g : Σ B → X be two maps,
Σ Σ Σp A B AA : ×( ) → , Σ Σ Σp A B BB : ×( ) →
the suspension of the corresponding projections and
′ = °f f pAΣ , ′ = °g g pBΣ ,
the composites respectively. Using the co-multiplication of Σ A B×( ), we have a
well-defined map
f g f g A B X′ ° ′( ) ° ′ ° ′( ) ×( ) →− −1 1 : Σ .
This map, when restricted to Σ A ∨ Σ B, is homotopic to the constant map. Now we let
K : Σ A B×( ) → X be a map homotopic to f g′ ° ′( )− −1 1 ° ′ ° ′( )f g whose restriction
to Σ A ∨ Σ B is the constant map.
Definition 4.1. The map K : Σ A B×( ) → X , as above, defines a map K ′ :
Σ A B×( ) → X and the homotopy class ′[ ]K is a well-defined class called the
generalized Whitehead product of f[ ] a n d g[ ], and is denoted by f[ ] ° g[ ]
(see [3]).
In the spirit of [2], we reinterpret Arkowitz’s generalized Whitehead product as
follows.
Theorem 4.1. Given α ∈[ ]Σ A X, and β ∈[ ]Σ B X, , then the image of α β°( )
in τA B X× ( ) is the commutator of the image of α−1 and the image of β−1 i n
τA B X× ( ).
Proof. The image of f[ ] ° g[ ] in τA B X× ( ) is the homotopy class of the
composite
Σ Σ ΣA B A B A B X×( )( ) → ×( ) → ∧( ) →°∪ *
α β
.
Call f[ ], g[ ] the images of f[ ], g[ ] in τA B X× ( ), respectively, which are the
homotopy classes of the composites
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GENERALIZATIONS OF FOX HOMOTOPY GROUPS, WHITEHEAD PRODUCTS … 327
Σ Σ ΣA B A B A X
f
×( )( ) → ×( ) → →∪ *
and
Σ Σ ΣA B A B B X
g
×( )( ) → ×( ) → →∪ * ,
respectively. The definition of the operation in τA B X× ( ) uses the fact that the domain
Σ A B×( )( )∪ * is a suspension. The commutator of the images of α−1 and β−1 in
Σ A B X×( )[ ], , by the definition of the operation when the domain is a suspension, has
the map f g′ ° ′( )− −1 1 ° ′ ° ′( )f g as a representative. But the homotopy class of this
map is also from (see [3]) the image of the generalized Whitehead product in
Σ A B X×( )[ ], . Now by composing with the suspension of the projection
Σ A B×( )( )∪ * → Σ A B×( ), the result follows.
Extending the definition of the classical Gottlieb group, we can define the
generalized Gottlieb group G ΣV X,( ) as follows.
Definition 4.2. For any connected spaces A and X, with a basepoint x X0 ∈ ,
we define
G ΣA X,( ) : =
Im : , , , ,e A X A X xX
Xv* Σ Σ1 0( )[ ] → ( )[ ]( )
to be the generalized Gottlieb group.
Similarly, we let
˜ ,G ΣA X( ) : = α: ΣA X F j a X→ ° ≈ ∨{ 1 for some F A X X: Σ × → },
where j : Σ A ∨ X ⊂→ Σ A × X is the inclusion.
The following result follows from [14].
Proposition 4.1. Suppose A and X are compactly generated
G ΣA X,( ) =
˜ ,G ΣA X( ).
We now analyze the question “Is G central in some group?”. In [14], it was
shown that for a co-H-space A, the Gottlieb group ˜ ,G A X( ) is central in A X,[ ].
(See also [15] for related results.) In our case, as a result of the projection V × W → V,
we can regard ΣV X,[ ] as a subgroup of Σ V W X×( )[ ], . Under this identification, we
have the following theorem.
Theorem 4.2. The generalized Gottlieb group G ΣV X,( ), regarded as a
subgroup of τV W X× ( ) , is central in τV W X× ( ) for any W . In particular, it is
central in ΣV X,[ ].
Proof. Let us consider the image of an element of ΣV X,[ ] in Σ V W X×( )[ ],
under the composite
Σ ΣΣ
V W V X
p f
×( ) → → .
Since we assume that f belongs to G ΣV X,( ), the map
f V X XX∨ ∨ →1 : Σ
has an extension H : Σ V × X → X, and hence the map
f p V W X XX°( ) ∨ ×( ) ∨ →Σ Σ1 :
has an extension to the product Σ V W X×( ) × .
Now, consider the composite
Σ Σ ΣV W V W V W X X
g H×( ) × ×( ) → ×( ) × →− ×1
,
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328 M. GOLASINSKI′ , D. GONÇALVES, P. WONG
where g is an arbitrary map from Σ V W×( ) to X. We use the fact that
τ τ τV W V W V WX Y X Y× × ××( ) ≅ ×( ) ( )
as groups so that an arbitrary element from τV W X× ( ) commutes with an arbitrary
element of τV W Y× ( ), when they are regarded as elements of τV W X Y× ×( ). If we
apply the above observation for our case, we obtain that the images of g[ ] and of
f p°[ ]Σ in τV W X× ( ) commute and the result follows.
As a consequence of Theorem 4.1, which says that in the group τV W X× ( ) the
generalized Whitehead product becomes a commutator, and of the Theorem above, we
obtain that the Whitehead product of an element with any element of the Gottlieb group
vanishes.
By combining Theorem 4.1 and Theorem 4.2, we deduce the following result first
obtained by K. Varadarajan [4].
Theorem 4.3. Given α ∈ ( )G ΣA X, , for any β ∈[ ]ΣB X, , we have
α β° = 0.
When the target space X is a suspension, an immediate consequence of the result
above gives the following description of the generalized Gottlieb groups.
Corollary 4.1. Let
P Σ Σ Σ Σ Σ ΣV X V X V X X, : : , ,( ) = [ ] → ∧( )[ ]( )Ker ω ,
where ω is given by the generalized Whitehead product. Then,
G PΣ Σ Σ ΣV X V X, ,( ) = ( ) .
Remark 4.1. The equality of Theorem 4.3 was also shown by C. Hoo [15] and
later generalized by H. Marcum [16].
Remark 4.2. Given a fibration F ⊂→ E → B, there is an associated Eckmann –
Hilton exact sequence with a boundary map ∂ : A B, Ω[ ] → A F,[ ] for any locally
finite CW complex A. In [4], it was shown that ∂ [ ]( )A B, Ω ⊆ G A F,( ). Thus, when
A = Σ T n−( )1 ∪ * , Theorem 2.2 is in fact a special case of Varadarajan’s result.
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Received 15.07.2004
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