Cyclic and cocyclic maps and generalized Whitehead products
Given co-H-spaces $X$ and $Y$, B. Gray has defined a co-H-space $X\circ Y$ and a natural transformation$X\circ Y\to X\vee Y$ which leads to a generalized Whitehead product. We make use of that product and sketch ideas on its dual to examine cyclic and cocyclic maps. Given spaces $X$ and...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552771035594752 |
|---|---|
| author | Golasiński, M. de Melo, T. Golasiński, M. de Melo, T. |
| author_facet | Golasiński, M. de Melo, T. Golasiński, M. de Melo, T. |
| author_institution_txt_mv | [
{
"author": "M. Golasiński",
"institution": "University of Warmia and Mazury"
},
{
"author": "T. de Melo",
"institution": "Instituto de Geociências e Ciências Exatas"
}
] |
| author_sort | Golasiński, M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | Given co-H-spaces $X$ and $Y$, B. Gray has defined a co-H-space $X\circ Y$ and a natural transformation$X\circ Y\to X\vee Y$ which leads to a generalized Whitehead product. We make use of that product and sketch ideas on its dual to examine cyclic and cocyclic maps. Given spaces $X$ and $Y$, some results on Gottlieb sets $\mathcal{G}(X,Y)$ and dual Gottlieb sets $\mathcal{DG}(X,Y)$ are stated. |
| first_indexed | 2026-08-04T01:04:43Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 22–34
Marek Golasiński, Thiago de Melo
(Faculty of Mathematics and Computer Science
University of Warmia and Mazury
S loneczna 54, 10-710 Olsztyn, Poland, Instituto de Geociências e
Ciências Exatas
UNESP–Univ Estadual Paulista
Av. 24A, 1515, Bela Vista. CEP 13.506–900. Rio Claro–SP, Brazil)
Cyclic and cocyclic maps and
generalized Whitehead products
marekg@matman.uwm.edu.pl, tmelo@rc.unesp.br
Given co-H-spaces X and Y , B. Gray [13] has defined a co-H-space X ◦ Y
and a natural transformation X ◦ Y → X ∨ Y which leads to a generalized
Whitehead product. We make use of that product and sketch ideas on
its dual to examine cyclic and cocyclic maps. Given spaces X and Y ,
some results on Gottlieb sets G(X,Y ) and dual Gottlieb sets DG(X,Y ) are
stated.
Introduction
The Gottlieb group Gn(X) of a space X is the subgroup of the
homotopy group πn(X) of X consisting of homotopy classes of maps
f : Sn → X such that the map f ∨ idX : Sn ∨ X → X admits an ex-
tension F : Sn × X → X. The study of the properties and structure
of the Gottlieb groups represents a fundamental problem in homotopy
theory dating back to their introduction by D. Gottlieb in the 1960’s
c© Marek Golasiński, Thiago de Melo, 2013
Marek Golasiński, Thiago de Melo 23
[8, 10]. Connections between the Gottlieb groups and fixed point the-
ory [8, 15, 22], transformation groups [11, 20], covering spaces [11, 16]
and the homotopy theory of fibrations [9, 12, 21] have been extensively
researched.
The definition of Gn(X) uses the concept of cyclic homotopies. K.
Varadarajan [23] studies the role of cyclic and cocyclic (dual of cyclic)
maps in the set-up of Eckmann-Hilton duality. The set of homotopy
classes of cyclic maps X → Y , denoted by G(X,Y ) is a group provided
X carries an H-cogroup structure. Dually, the set of homotopy classes
of cocyclic maps X → Y , denoted by DG(X,Y ) is a group provided Y
carries an H-group structure. Relationships between these generalized
Gottlieb (dual Gottlieb groups) and the generalized Whitehead product
(the dual generalized Whitehead product) [1] have been considered in
[14, 17, 18, 19] and other various papers.
The aim of this paper is to present those results in the context of
the so called Theriault product considered by B. Gray in [13] being an
extended version of the generalized Whitehead product from [1] and its
dual. The first section expounds the notions and clarify results needed in
next two sections. Section 2 recalls results on cyclic maps and then takes
up the systematic study of these maps in the context of results from [13].
Section 3 is devoted to cocyclic maps. First, their relations with the
dual generalized Whitehead product [1] are summarized. In particular,
a characterization of co-H-spaces in terms of the cocyclicity of maps is
concluded. Then, following mutatis mutandis the construction presented
by B. Gray in [13] and the cotelescope concept, we sketch ideas of the
dual Theriault product extending the dual generalized Whitehead [1]
and relate cocyclic maps to this product. Many results and proofs on
the Theriault product can be dualized. The details will be published
somewhere shortly.
Acknowledgements. This work was started during the visit of the first
author to the Instituto de Geociências e Ciências Exatas, UNESP–Univ
Estadual Paulista, Rio Claro–SP (Brazil) in the period from August 17–
27, 2012. He would like to thank that Institute for its hospitality and
24 Cyclic and cocyclic maps and generalized Whitehead products
supporting during his stay.
2 Prerequisites
We concentrate with connected and based spaces having the homo-
topy type of CW -complexes. All maps and homotopies preserve base
points. For simplicity, we sometimes use the same symbol for a map
and its homotopy class. Denote by [X,Y ] the set of homotopy classes of
continuous maps X → Y and write Sn for the n-dimensional sphere. In
particular, let πn(X) = [Sn, X] be the nth homotopy group of a space X
for n ≥ 0.
Next, write ΣX and ΩX for the suspension and the loop space of X.
Recall that ΣX and ΩX are an H-cogroup and an H-group, respectively.
If f : X → Y then for every space Z, we have homomorphisms (Σf)∗ :
[ΣY, Z] → [ΣX,Z] and (Ωf)∗ : [Z,ΩX] → [Z,ΩY ]. Further, there are
canonical natural maps e : ΣΩX → X and e′ : X → ΩΣX.
The following well-known results are frequently used:
Proposition 2.1. (1) If X is a co-H-space, then there is a map s : X →
ΣΩX such that es ' idX ;
(2) If X is an H-space, then there is a map s′ : ΩΣX → X such that
s′e′ ' idX ;
(3) Let X and Y be an H-cogroup and an H-group, respectively. Then,
[X,Z] and [Z, Y ] are groups for any space Z.
Let X[Y be the flat product and X ∧ Y the smash product, that is,
the fibre and the cofibre of the inclusion X ∨ Y ↪→ X × Y . Next, write
∆ : X → X ×X and ∇ : X ∨X → X for the diagonal and folding maps,
respectively.
The Whitehead product [−,−] : πm(X) × πn(X) → πm+n−1(X),
determined by the Whitehead map w : Sm+n−1 → Sm ∨ Sn plays a
crucial role in the homotopy theory. The generalized Whitehead map
w : Σ(X ∧ Y ) → ΣX ∨ ΣY constructed in [1] leads to the generalized
Whitehead product
[−,−] : [ΣX,Z]× [ΣY,Z]→ [Σ(X ∧ Y ), Z].
Marek Golasiński, Thiago de Melo 25
Now, let CO be the category of simply connected co-H-spaces and co-H-
maps. In [13], a functor
◦ : CO × CO → CO
(called the Theriault product) and a natural transformation w : X ◦Y →
X ∨ Y for co-H-spaces X,Y generalizing the Whitehead product have
been defined. More precisely, in [13, Theorem 1, Theorem 2] it has been
shown:
Theorem 2.2. There is a functor
◦ : CO × CO −→ CO
and equivalences in CO:
(1) (ΣX) ◦ Y ∼= X ∧ Y ;
(2) Σ(X ◦ Y ) ∼= X ∧ Y ;
(3) (X1 ∨X2) ◦ Y ∼= (X1 ◦ Y ) ∨ (X2 ◦ Y )
and homotopy equivalences:
(4) X ◦ Y ∼= Y ◦X;
(5) (X ◦ Y ) ◦ Z ∼= X ◦ (Y ◦ Z).
Theorem 2.3. There is a natural transformation
w◦ : X ◦ Y −→ X ∨ Y
which is the Whitehead product map in case X and Y are both suspen-
sions. Furthermore, there is a homotopy equivalence
X × Y ∼= (X ∨ Y ) ∪w◦ C(X ◦ Y ),
where (X∨Y )∪w◦C(X ◦Y ) is the mapping cone of w◦ : X ◦Y −→ X∨Y .
Notice that w◦ : X ◦ Y −→ X ∨ Y defines a map
[−,−]◦ : [X,Z]× [Y,Z]→ [X ◦ Y, Z]
for any space Z.
26 Cyclic and cocyclic maps and generalized Whitehead products
3 Cyclic maps and evaluation groups
According to [23], a map f : X → Y is said to be cyclic if there exists
a map F : X × Y → Y such that the diagram
X ∨ Y� _
��
∇(f∨idY ) // Y
X × Y
F
88
is homotopy commutative.
Write G(X,Y ) for the set of homotopy classes of cyclic maps from X
to Y called the Gottlieb subset of [X,Y ]. If X is an H-cogroup then by
[23, Theorem 1.5] the subset G(X,Y ) ⊆ [X,Y ] is a subgroup of [X,Y ]. If
X = Sn, the n-dimensional sphere then G(Sn, Y ) = Gn(Y ) is called the
nth evaluation subgroup of Y or the nth Gottlieb group defined in [8] for
n = 1 and then in [10] for any n ≥ 1. Then, Gn+k(Sn) and Gn+k(FPn)
have been extensively studied in [6] and [7], respectively, where FPn
is the projective space over F being the reals R, complex numbers C,
quaternions H or the Cayley algebra K.
To show the existence of cyclic maps, we recall:
Proposition 3.1 ([23, Lemmas 1.3 and 1.4]). Let f : X → Y be a cyclic
map and g : Z → X an arbitrary map. Then:
(1) fg : Z → Y is a cyclic map;
(2) if a map g : Y → Y ′ has a right homotopy inverse then gf : X →
Y ′ is a cyclic map.
In particular, let X be a co-H-space, f : X → Y and e : ΣΩX → X
the usual map. Then f is cyclic if and only if fe : ΣΩX → Y is cyclic.
Proposition 3.2 ([17, Proposition 3.3]). Let Y be a space. Then the
following are equivalent:
(1) Y is an H-space;
(2) idY is cyclic;
(3) G(X,Y ) = [X,Y ] for any space X.
Marek Golasiński, Thiago de Melo 27
Another way in which cyclic maps arise naturally is by fibrations.
Suppose F → E → B is a fibration. Then we have an operation ρ :
F × ΩB → F and the restriction ∂ = ρ|ΩB is cyclic.
Now, we make use of Theorem 2.3 to deduce results being key ones in
sequel.
Corollary 3.3. Let X,Y be spaces. Then:
(1) the map w◦ : ΣΩX ◦ ΣΩY → ΣΩX ∨ ΣΩY coincides with the
generalized Whitehead map w : Σ(ΩX ∧ ΩY )→ ΣΩX ∨ ΣΩY ;
(2) there is the commutative diagram
X ◦ Y w◦ // X ∨ Y
ΣΩX ◦ ΣΩY
e◦e
OO
w◦ // ΣΩX ∨ ΣΩY.
e∨e
OO
Then, the result [18, Proposition 4.6] leads to:
Proposition 3.4. Let X be a co-H-space and f : X → Y a cyclic map.
Then [f, g]◦ = 0 for any map g : Z → Y provided Z is a co-H-space.
Proof. Let f : X → Y be a cyclic map. Then by Proposition 3.1 the
map fe : ΣΩX → Y is cyclic as well. Hence, in view of [18, Proposition
4.6], we get [fe, ge] = 0. Because X and Z are co-H-spaces, Corollary 3.3
leads to [f, g]◦ = 0 and the proof is complete.
Further [5, Proposition 2.3] and Proposition 2.1 yield:
Proposition 3.5. For a map f : X → Y of H-groups, the following are
equivalent:
(1) f∗ maps [Z,X] into the center of [Z, Y ];
(2) ∇(f ∨ idY )i ' ?, where i : X[Y ↪→ X ∨ Y is the inclusion map.
If one of the conditions above is fulfilled, T. Ganea [5] says that f
maps X into the center of Y .
The proof of the result below is a direct consequence of Corollary 3.3
and [14, Corollary 3].
28 Cyclic and cocyclic maps and generalized Whitehead products
Theorem 3.6. Let X,Y be co-H-spaces and f : X → Y . Then the
following are equivalent:
(1) f is cyclic;
(2) f maps ΩX into the center of ΩY ;
(3) [f, idY ]◦ = 0.
Theorem 3.6 generalized the result known to spheres: f ∈
G(Sn+k,Sn) = Gn+k(Sn) if and only if the Whitehead product [f, idSn ] = 0
which has been applied in [6] to find Gn+k(Sn) for k ≤ 13. Certainly, the
computations depend on the Whitehead product on spheres.
Now, let i1 : Y1 ↪→ Y1 ∨ Y2 and i2 : Y2 ↪→ Y1 ∨ Y2 be the inclusion
maps. Then, Theorem 3.6 leads to the following generalization of [3,
Proposition 2.3]:
Corollary 3.7. Let X,Y1, Y2 be co-H-spaces and f : X → Y1∨Y2. Then,
f is cyclic if and only if [f, i1]◦ = [f, i2]◦ = 0.
If A is an abelian group and n ≥ 2 then the Moore space M(A,n) is
a co-H-space as a suspension of some space. Because M(A1 ⊕ A2, n) ∼=
M(A1, n)∨M(A2, n) for some abelian groups A1, A2 [3, Proposition 2.3]
has been applied to compute Gn(M(A,n)) provided A is a finitely gener-
ated abelian group. The paper [2] considers the set of homotopy classes
of co-structures on a Moore space M(A,n), where A is an abelian group
and n ≥ 2 is an integer. It is shown that for n > 2 the set has one element
and for n = 2 the set is in one-to-one correspondence with Ext(A,A⊗A).
Further, a detailed investigation of the co-H-structures onM(A, 2) in the
case A = Zm, the integers mod m has been considered. It has been
shown that all co-H-structures on M(Zm, 2) are associative and commu-
tative ifm is odd, and all co-H-structures onM(Zm, 2) are associative and
non-commutative if m is even. Therefore, Corollary 3.7 should be use-
ful to describe G2(M(A, 2)) with respect to all possible co-H-structures
on M(A, 2) provided A is a finitely generated group or more generally,
A =
⊕
i∈I Z⊕
⊕
j∈J Zmj
.
Let Y be an H-group and f : X → Y . Recall that f is called central
if c(idY ×f) ' ?, where c : Y × Y → Y is the basic commutator map. If
Marek Golasiński, Thiago de Melo 29
Y is an H-space then, in view of Proposition 2.1, the map Ω : [X,Y ] →
[ΩX,ΩY ] given by f 7→ Ωf is injective. Write [ΩX,ΩY ]CΩ for the subset
of [ΩX,ΩY ] consisting of those homotopy classes of maps Ωf which are
central. Following [18, Definition 4.1], we set C(X,Y ) = Ω−1[ΩX,ΩY ]CΩ.
By [18, Propositions 4.6 and 5.1], it holds:
Proposition 3.8. Let X,Y and Z be spaces.
(1) If f ∈ C(ΣX,Z) then [f, g] = 0 for any g ∈ [ΣY,Z].
(2) C(X,Y ) is a subgroup contained in the center of [X,Y ] if X is a
co-H-space with a right homotopy inverse and Y is any space.
It follows that ifX is a co-H-space with a right homotopy inverse, then
for every space Y , G(X,Y ) ⊆ C(X,Y ) ⊆ center of [X,Y ] as subgroups.
In particular, G(X,Y ) and C(X,Y ) are abelian groups provided X is a
co-H-space. This generalizes Gottlieb’s result from [8] that the Gottlieb
group G1(Y ) lies in the center of the homotopy group π1(Y ).
4 Cocyclic maps and coevaluation groups
According to [23], a map f : X → Y is said to be cocyclic if there is
a map F ′ : X → X ∨ Y such that the diagram
X × Y
X
(idX ×f)∆
99
F ′
// X ∨ Y
?�
OO
is homotopy commutative.
Write DG(X,Y ) for the set of homotopy classes of cocyclic maps from
X to Y called the dual Gottlieb subset of [X,Y ]. If Y is an H-group
then by [23, Theorem 1.5] the subset DG(X,Y ) ⊆ [X,Y ] is a subgroup
of [X,Y ].
Certainly, every map f : X → Y is cocyclic provided X is a co-H-
space.
30 Cyclic and cocyclic maps and generalized Whitehead products
Another way in which cocyclic maps arise naturally is by cofibrations
(cf. [19]). Suppose A → B → C is a cofibration. Then we have a
cooperation φ : C → C ∨ ΣA. Then the map s = p2φ : C → ΣA is
cocyclic, where p2 : C ∨ ΣA→ ΣA is the projection map.
Notice that if f : X → Y is a cocyclic map and g : X ′ → X has a
left homotopy inverse then fg : X ′ → Y is also a cocyclic map. Then, in
view of [23, Lemma 7.2], Proposition 3.1 can be dualized as follows:
Proposition 4.1. Let f : X → Y be a cocyclic map. Then:
(1) gf : X → Z is a cocyclic map for an arbitrary map g : Y → Z;
(2) if a map g : X ′ → X has a left homotopy inverse then fg : X ′ → Y
is a cocyclic map.
In particular, let Y be an H-space, f : X → Y and e′ : Y → ΩΣY the
usual map. Then f is cocyclic if and only if e′f : X → ΩΣY is cocyclic.
Further, [19, Proposition 3.2] provides a characterization of a co-H-space
in terms of the cocyclicity of maps.
Proposition 4.2. Let X be a space. Then the following are equivalent:
(1) X is a co-H-space;
(2) idX is cocyclic;
(3) DG(X,Y ) = [X,Y ] for any space Y .
Recall from [1] that given spaces X and Y , there is a dual Whitehead
map w′ : ΩX × ΩY → Ω(X[Y ). This leads to the dual generalized
Whitehead product
[−,−]′ : [Z,ΩX]× [Z,ΩY ]→ [Z,Ω(X[Y )]
for any space Z.
Now, let CO′ be the category of simply connected H-spaces and H-
maps. Following mutatis mutandis the construction presented by B. Gray
in [13] and the cotelescope construction, we get a functor
◦′ : CO′ × CO′ −→ CO′
(called the dual Theriault product) and a natural transformation
w′ : X × Y −→ X ◦′ Y
Marek Golasiński, Thiago de Melo 31
which leads to a map
[−,−]◦′ : [Z,X]× [Z, Y ]→ [Z,X ◦′ Y ]
for H-spaces X,Y and any space Z. Many results and proofs of [−,−]◦
can be dualized. We mention only that the products [−,−]′ and [−,−]◦′
coincide provided X,Y are loop spaces. However, many cannot since
[−,−]◦′ is not precise a dual of [−,−]◦. The details and dual version of
Theorem 2.2 and Theorem 2.3 will be published somewhere shortly.
The dual version of Corollary 3.3 and the result [18, Proposition 4.6]
yield:
Proposition 4.3. Let Y be an H-space and f : X → Y a cocyclic map.
Then [f, g]◦′ = 0 for any map g : X → Z provided Z is an H-space.
>From this a dual version of Corollary 3.7 follows:
Corollary 4.4. Let X1, X2, Y be H-spaces and f : X1×X2 → Y . Then,
f is cocyclic if and only if [f, p1]◦′ = [f, p2]◦′ = 0 for the projection maps
p1 : X1 ×X2 → X1 and p2 : X1 ×X2 → X2.
Let A be an abelian group and n ≥ 2. Then the associated
Eilenberg-MacLane space K(A,n) inherits an H-structure. Because
K(A1 × A2, n) ∼= K(A1, n) × K(A2, n) for any abelian groups A1, A2,
Corollary 4.4 should be very useful to compute DG(K(A,n), Y ) provided
that A is an abelian finitely generated group and Y is an H-space.
The dual version of Proposition 3.5 and [5, Proposition 2.3] lead to:
Proposition 4.5. For a map f : X → Y of H-cogroups, the following
are equivalent:
(1) f∗ maps [Y, Z] into the center of [X,Z];
(2) j(idX ×f)∆ ' ?, where j : X × Y → X ∧ Y is the quotient map.
If one of the conditions above is fulfilled, we follow T. Ganea [5] to
say that f maps X into the cocenter of Y . Let X be an H-cogroup and
f : X → Y . Recall that f is called cocentral if (idX ∨f)c ' ?, where
c : X → X ∨X is the basic cocommutator map.
32 Cyclic and cocyclic maps and generalized Whitehead products
If X is a co-H-space then the map Σ : [X,Y ] → [ΣX,ΣY ] given by
f 7→ Σf is injective. A subset DC(X,Y ) of [X,Y ] which is the dual of
C(X,Y ) has been studied in [19]. If Y is an H-space then the map Σ :
[X,Y ]→ [ΣX,ΣY ] given by f 7→ Σf is injective. Let [ΣX,ΣY ]CΣ denote
the subset of [ΣX,ΣY ] consisting of those homotopy classes of maps Σf
which are cocentral. Following [19, Definition 4.7], we set DC(X,Y ) =
Σ−1[ΣX,ΣY ]CΣ.
In view of [19, Propositions 4.8 and 5.2], it holds:
Proposition 4.6. Let X,Y and Z be spaces.
(1) If f ∈ DC(Z,ΩX) then [f, g]′ = 0 for any g ∈ [Z,ΩY ];
(2) the set DC(X,Y ) is a subgroup contained in the center of [X,Y ]
if Y is an H-space with a left homotopy inverse and X is any space.
It follows that if Y is an H-space with a right homotopy inverse,
then for every space X there are inclusions DG(X,Y ) ⊆ DC(X,Y ) ⊆
center of [X,Y ] of subgroups. In particular, DG(X,Y ) and DC(X,Y )
are abelian groups provided X is an H-space.
References
[1] M. Arkowitz, The generalized Whitehead product, Pacific J. Math.
12 (1962), 7–23.
[2] M. Arkowitz, M. Golasiński, Co-H-structures on Moore spaces of
type (G, 2), Canad. J. Math. 46 (1994), 673–686.
[3] M. Arkowitz, K.-I. Maruyama, The Gottlieb group of a wedge of
suspensions, (preprint).
[4] W. D. Barcus, M. G. Barratt, On the homotopy classification of the
extensions of a fixed map, Trans. Amer. Math. Soc. 88 (1958), 57–74.
[5] T. Ganea, Induced fibrations and cofibrations, Trans. Amer. Math.
Soc. 127 (1967), 442–459.
[6] M. Golasiński, J. Mukai, Gottlieb groups of spheres, Topology 47
(2008), 399–430.
Marek Golasiński, Thiago de Melo 33
[7] M. Golasiński, J. Mukai, Gottlieb and Whitehead center groups of
projective spaces, (submitted).
[8] D. H. Gottlieb, A certain subgroup of the fundamental group, Amer.
J. Math. 87 (1965), 840–856.
[9] D. H. Gottlieb, On fibre spaces and the evaluation map, Ann. of
Math. 87 (1968), 42–55.
[10] D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J.
Math. 91 (1969), 729–756.
[11] D. H. Gottlieb, Covering transformations and universal fibrations,
Illinois J. Math. 13 (1969), 432–437.
[12] D. H. Gottlieb, Applications of bundle map theory, Trans. Amer.
Math. Soc. 171 (1972), 23–50.
[13] B. Gray, On generalized Whitehead products, Trans. Amer. Math.
Soc. 11 (2011), 6143–6158.
[14] C. S. Hoo, Cyclic maps from suspensions to suspensions, Canad. J.
Math. 24 (1972), 789–791.
[15] B.-J. Jiang, Estimation of the Nielsen numbers, Acta Math. Sinica
14 (1964), 330–339.
[16] G. E. Lang, Evaluation subgroups of factor spaces, Pacific J. Math.
42 (1972), 701–709.
[17] K. L. Lim, On cyclic maps, J. Austral. Math. Soc. Ser. A 32 (1982),
349–357.
[18] K. L. Lim, On evaluation subgroups of generalized homotopy groups,
Canad. Math. Bull. 27 (1) (1984), 78–86.
[19] K. L. Lim, Cocyclic maps and coevaluation subgroups, Canad. Math.
Bull. 30 (1) (1987), 63–71.
34 Cyclic and cocyclic maps and generalized Whitehead products
[20] G. Lupton, J. Oprea, Cohomologically symplectic spaces: toral ac-
tions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1)
(1995), 261–288.
[21] J. Oprea, The Samelson space of a fibration, Michigan Math. J. 34
(1) (1987), 127–141.
[22] J. Oprea, Gottlieb groups, group actions, fixed points and rational
homotopy, Lecture Notes Series 29, Seoul National University, Re-
search Institute of Mathematics, Global Analysis Research Center,
Seoul, 1995.
[23] K. Varadarajan, Generalized Gottlieb groups, J. Indian Math. Soc.
33 (1969), 141–164.
|
| id | oai:trim.imath.kiev.ua:article-273 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:04:43Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/11/37c8abcfa39c98754faa8a84f6da0c11.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2732018-02-10T20:56:26Z Cyclic and cocyclic maps and generalized Whitehead products Циклічні та коциклічні відображення і узагальнені добутки Уайтхеда Golasiński, M. de Melo, T. Golasiński, M. de Melo, T. Given co-H-spaces $X$ and $Y$, B. Gray has defined a co-H-space $X\circ Y$ and a natural transformation$X\circ Y\to X\vee Y$ which leads to a generalized Whitehead product. We make use of that product and sketch ideas on its dual to examine cyclic and cocyclic maps. Given spaces $X$ and $Y$, some results on Gottlieb sets $\mathcal{G}(X,Y)$ and dual Gottlieb sets $\mathcal{DG}(X,Y)$ are stated. Дано ко-H-простір $X$ і $Y$. Б.Грей визначив ко-H-простір $X\circ Y$  і природне перетворення$X\circ Y\to X\vee Y$, яке приводить до  узагальненого добутку Уайтхеда. Ми використовуємо цей добуток та  накреслюємо ідею його двоїстого добутку для вивчення циклічних та коциклічних відображень.  Стверджуються деякі рузультати для множин Готліба $\mathcal{G}(X,Y)$ та двоїстих множин Готліба $\mathcal{DG}(X,Y)$ для просторів  $X$ та $Y$. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/273 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 22-34 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 22-34 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 22-34 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/273/278 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Golasiński, M. de Melo, T. Golasiński, M. de Melo, T. Cyclic and cocyclic maps and generalized Whitehead products |
| title | Cyclic and cocyclic maps and generalized Whitehead products |
| title_alt | Циклічні та коциклічні відображення і узагальнені добутки Уайтхеда |
| title_full | Cyclic and cocyclic maps and generalized Whitehead products |
| title_fullStr | Cyclic and cocyclic maps and generalized Whitehead products |
| title_full_unstemmed | Cyclic and cocyclic maps and generalized Whitehead products |
| title_short | Cyclic and cocyclic maps and generalized Whitehead products |
| title_sort | cyclic and cocyclic maps and generalized whitehead products |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/273 |
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