Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13
Saved in:
| Published in: | Український математичний журнал |
|---|---|
| Date: | 2013 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2013
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/165595 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 / M. Golasinski, T. de Melo // Український математичний журнал. — 2013. — Т. 65, № 8. — С. 1023-1034. — Бібліогр.: 21 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-165595 |
|---|---|
| record_format |
dspace |
| spelling |
Golasiński, M. de Melo, T. 2020-02-14T12:01:20Z 2020-02-14T12:01:20Z 2013 Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 / M. Golasinski, T. de Melo // Український математичний журнал. — 2013. — Т. 65, № 8. — С. 1023-1034. — Бібліогр.: 21 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165595 517.98 The second author was partially supported by: PROPG–UNESP and FAPESP 2012/07301-3 en Інститут математики НАН України Український математичний журнал Статті Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 Оціночні розшарування i компоненти лiнiйної зв'язності простору відображень M(Sⁿ⁺ᵏ,Sⁿ) при 8 ≤ k ≤ 13 Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 |
| spellingShingle |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 Golasiński, M. de Melo, T. Статті |
| title_short |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 |
| title_full |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 |
| title_fullStr |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 |
| title_full_unstemmed |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 |
| title_sort |
evaluation fibrations and path-components of the mapping space m(sⁿ⁺ᵏ,sⁿ) for 8 ≤ k ≤ 13 |
| author |
Golasiński, M. de Melo, T. |
| author_facet |
Golasiński, M. de Melo, T. |
| topic |
Статті |
| topic_facet |
Статті |
| publishDate |
2013 |
| language |
English |
| container_title |
Український математичний журнал |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Оціночні розшарування i компоненти лiнiйної зв'язності простору відображень M(Sⁿ⁺ᵏ,Sⁿ) при 8 ≤ k ≤ 13 |
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165595 |
| citation_txt |
Evaluation Fibrations and Path-Components of the Mapping Space M(Sⁿ⁺ᵏ,Sⁿ) for 8 ≤ k ≤ 13 / M. Golasinski, T. de Melo // Український математичний журнал. — 2013. — Т. 65, № 8. — С. 1023-1034. — Бібліогр.: 21 назв. — англ. |
| work_keys_str_mv |
AT golasinskim evaluationfibrationsandpathcomponentsofthemappingspacemsnksnfor8k13 AT demelot evaluationfibrationsandpathcomponentsofthemappingspacemsnksnfor8k13 AT golasinskim ocínočnírozšaruvannâikomponentiliniinoízvâznostíprostoruvídobraženʹmsnksnpri8k13 AT demelot ocínočnírozšaruvannâikomponentiliniinoízvâznostíprostoruvídobraženʹmsnksnpri8k13 |
| first_indexed |
2025-11-25T20:56:27Z |
| last_indexed |
2025-11-25T20:56:27Z |
| _version_ |
1850543681210155008 |
| fulltext |
UDC 517.98
M. Golasiński (Univ. Warmia and Mazury, Poland),
Thiago de Melo (Inst. de Geociências e Ciências Exatas, Brazil)
EVALUATION FIBRATIONS AND PATH-COMPONENTS
OF THE MAPPING SPACE M(Sn+k, Sn) FOR 8 ≤ k ≤ 13*
ОЦIНОЧНI РОЗШАРУВАННЯ I КОМПОНЕНТИ ЛIНIЙНОЇ ЗВ’ЯЗНОСТI
ПРОСТОРУ ВIДОБРАЖЕНЬ M(Sn+k, Sn) ПРИ 8 ≤ k ≤ 13
Let M(Sm, Sn) be the space of maps from the m-sphere Sm into the n-sphere Sn with m,n ≥ 1. We estimate the
number of homotopy types of path-components Mα(Sn+k, Sn) and fiber homotopy types of the evaluation fibrations
ωα : Mα(Sn+k, Sn) → Sn for 8 ≤ k ≤ 13 and α ∈ πn+k(Sn) extending the results of [Mat. Stud. – 2009. – 31, № 2. –
P. 189 – 194]. Further, the number of strong homotopy types of ωα : Mα(Sn+k, Sn) → Sn for 8 ≤ k ≤ 13 is determined
and some improvements of the results from [Mat. Stud. – 2009. – 31, № 2. – P. 189 – 194] are obtained.
Нехай M(Sm, Sn) — простiр вiдображень iз m-сфери Sm в n-сферу Sn з m,n ≥ 1. Ми оцiнюємо число типiв
гомотопiї для компонент лiнiйної зв’язностi Mα(Sn+k, Sn) та типiв гомотопiй шарiв для оцiночних розшарувань
ωα : Mα(Sn+k, Sn) → Sn при 8 ≤ k ≤ 13 та α ∈ πn+k(Sn), узагальнюючи результати з [Mat. Stud. – 2009. – 31,
№ 2. – P. 189 – 194]. Крiм того, визначаємо число типiв сильних гомотопiй ωα : Mα(Sn+k, Sn)→ Sn при 8 ≤ k ≤ 13
та отримуємо деякi покращення результатiв з [Mat. Stud. – 2009. – 31, № 2. – P. 189 – 194].
1. Introduction. Given spaces X and Y, let M(X,Y ) be the mapping space of all continuous maps
ofX into Y equipped with the compact-open topology. The spaceM(X,Y ) is generally disconnected
and its path-components are in one-to-one correspondence with the set [X,Y ] of (free) homotopy
classes of maps of X into Y.
Given x0 ∈ X, consider the evaluation map
ω : M(X,Y )→ Y
defined by ω(f) = f(x0) for f ∈ M(X,Y ). Let Mα(X,Y ) be the path-component of M(X,Y )
which contains all maps in α ∈ [X,Y ]. By [13, p. 83] (Theorem III.13.1), the evaluation map
ωα : Mα(X,Y ) → Y obtained by restricting ω to Mα(X,Y ) is a Hurewicz fibration provided X is
locally compact. Then, the natural classification problems arise:
(1) divide the set of path-components of M(X,Y ) into homotopy types;
(2) divide the set of evaluation fibrations ωα : Mα(X,Y ) → Y into fibre- and strong fibre-
homotopy types for α ∈ [X,Y ].
Conditions for when two path-components of M(X,Y ) are homotopy equivalent are presented
in [16] provided that spaces X and Y are connected and countable CW -complexes.
Let now Sn be the n-sphere. To study coincidences of fiberwise maps between sphere bundles
over S1, the set of fiberwise homotopy classes of those maps has been considered in [7]. But, the set
* The second author was partially supported by: PROPG–UNESP and FAPESP 2012/07301-3.
c© M. GOLASIŃSKI, THIAGO DE MELO, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8 1023
1024 M. GOLASIŃSKI, THIAGO DE MELO
of fiberwise maps between the trivial bundles S1 × Sm and S1 × Sn over S1 coincides with the free
loop space LM(Sm, Sn) = M(S1,M(Sm, Sn)).
Certainly, for the space M(Sm, Sn) with m,n ≥ 1, the path-components can be enumerated
by the homotopy group πm(Sn). In view of [10] (Theorem 4.1), there is a strong relation between
evaluation fibrations ωα : Mα(Sm,Sn)→ Sn for α ∈ πm(Sn) and the Whitehead product [ιn, α]. This
was used in [10] (Theorems 5.1, 5.2) to tackle a complete homotopy classification of path-components
of M(Sm,Sn) for m = n, n + 1 and compute the order of the homotopy group πn−1(Mα(Sn,Sn)).
Homotopy properties of various Mα(Sm,Sn) have been studied in [1, 14, 20].
The purpose of this note is to extend the results of [6] for m = n+ k with 8 ≤ k ≤ 13.
Section 1 summarizes [10, 11] and follows [16] to connect in Theorem 1.1 these classification
problems for M(Sm, Sn) with the m-th Gottlieb group Gm(Sn) considered in [8, 9] and then studied
in [5].
Section 2 makes use of [5] to take up the systematic study of the quotient sets πn+k(Sn)/ ±
± Gn+k(Sn) with 0 ≤ k ≤ 13. Then, our basic results stated in Propositions 2.1 – 2.6 estimate
the number of homotopy types of path-components of M(Sn+k,Sn) and fibre-homotopy types of
evaluation fibrations ωα : Mα(Sn+k, Sn)→ Sn with 0 ≤ k ≤ 13. Further, the number of strong fibre-
homotopy types of ωα : Mα(Sn+k,Sn)→ Sn with 0 ≤ k ≤ 13 is determined. Corollary 2.1 concludes
a list of evaluation fibrations ωα : Mα(Sn+k,Sn)→ Sn which are fibre-homotopy equivalent but not
strong fibre-homotopy equivalent for some 0 ≤ k ≤ 13.
Those results are applied in Section 3 to estimate the number of homotopy types of path-
components of M(S(n+1)d+k−1,FPn) and (strong) fibre-homotopy types of evaluation fibrations
ωα : Mα(S(n+1)d+k−1,FPn) → FPn with 0 ≤ k ≤ 13 for the n-projective spaces FPn for
F = R,C,H. Further, we deduce that path-components of M(Sm,KP 2) have the same homotopy
type for m ≤ 21, where KP 2 is the Cayley projective plane.
The last Section 4 makes use of [4, 18] to present the rational homotopy type of M(Sm, Sn) and
path-components of M(M(A,m),Sn) for a Moore spaceM(A,m).
1. Prerequisites. Given x0 ∈ X and y0 ∈ Y, write M(X,Y )∗ for the space of all continuous
pointed maps of X into Y. This leads to the Hurewicz fibration M(X,Y )∗ → M(X,Y )
ω→ Y,
provided X is locally compact. Recall that on the set [X,Y ]∗ of homotopy classes of pointed maps
there is an action of π1(Y, y0) and [X,Y ]∗/π1(Y, y0) = [X,Y ] [21] (Chapter I, (1.11)).
In particular, for π1(Y, y0) = 0 we get [X,Y ]∗ = [X,Y ], e.g., πm(Sn) = [Sm,Sn]∗ = [Sm,Sn],
for n > 1. Further, there is the Hurewicz fibration
M(Sm, Sn)∗ →M(Sm, Sn)
ω→ Sn.
A fibration p : E → B with a fibre F means a Hurewicz fibration together with a fixed homotopy
equivalence i : F → p−1(b0) over the base point b0 ∈ B. Recall that for fibrations p1 : E1 → B and
p2 : E2 → B a based map f : E1 → E2 is:
1) a fibre homotopy equivalence (fhe) if there exists g : E2 → E1 such that g ◦ f and f ◦ g are
homotopic to the respective identities by based homotopies F and G satisfying p1◦F (e1, t) = p1(e1)
and p2 ◦G(e2, t) = p2(e2) for e1 ∈ E1, e2 ∈ E2 and t ∈ [0, 1];
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
EVALUATION FIBRATIONS AND PATH-COMPONENTS OF THE MAPPING SPACE M(Sn+k, Sn) . . . 1025
2) a strong fibre homotopy equivalence (sfhe) if it is a fibre homotopy equivalence and i′2 ◦ f ◦ i1
is homotopic to the identity map idF , where i′2 is an arbitrary homotopy inverse of i2.
Let X be a connected and pointed space. The m-th Gottlieb group Gm(X) [8, 9] of a space X is
the subgroup of the m-th homotopy group πm(X) containing all elements which can be represented
by a map f : Sm → X such that f ∨ idX : Sm ∨ X → X extends (up to homotopy) to a map
F : Sm ×X → X. Observe that Gm(X) = πm(X) provided X is an H-space.
Given α ∈ πm(Sn) we have deduced in [5] that α ∈ Gm(Sn) if and only if the Whitehead product
[ιn, α] = 0, where ιn denotes the homotopy class of idSn . In other words, Gm(Sn) = ker [ιn,−] for
the map [ιn,−] : πm(Sn) → πm+n−1(Sn) with m ≥ 1. Write ]g for the order of the element g in a
group G. Then, by [5] (Section 2), from this interpretation of Gottlieb groups of spheres, we obtain
Gm(Sn) = (][ιn, α])πm(Sn),
if πm(Sn) is a cyclic group with a generator α. It follows thatGm(Sn) = πm(Sn) (resp.Gm(Sn) = 0)
provided ][ιn, α] = 1 (resp. ][ιn, α] =∞) for α ∈ πm(Sn). Furthermore, because of H-structures on
the spheres Sn for n = 1, 3, 7, it holds Gm(Sn) = πm(Sn) for any m ≥ 1.
Given a group G and its subgroup G′ < G, write G/±G′ for the quotient set of G by the relation
∼ defined as follows: for x, y ∈ G, x ∼ y if and only if xy ∈ G′ or xy−1 ∈ G′. Observe that if
G′i < Gi, i = 1, 2, then there is a surjection
φ : (G1 ×G2)/± (G′1 ×G′2)→ (G1/±G′1)× (G2/±G′2)
defined by (g1, g2) 7→ (g1, g2), which is not injective in general, where ḡ states for the appropriate
abstract class determined by g.
Example 1.1. (1) If G1 = G2 = Z and G′1 = G′2 = 3Z for the infinite cyclic group Z, then
|(Z×Z)/±(3Z×3Z)| = 5 and |Z/±3Z|2 = 4. Let Zn be the cyclic group with order n. If G1 = Z3,
G2 = Z6, G
′
1 = 0 and G′2 = Z2 then |(Z3 × Z6)/± (0× Z2)| = 5 and |Z3/± 0||Z4/± Z2| = 4.
(2) If G′1 = G1 then the bijection holds easily.
Writing ' for the homotopy equivalence relation, [10] (Theorems 1, 2) and [11] (Theorem 2.3)
lead to:
Theorem 1.1. Let m,n ≥ 1. Then, there are surjections:
πm(Sn)/±Gm(Sn) −→ {Mα(Sm, Sn); α ∈ πm(Sn)}/ ', (1.1)
πm(Sn)/±Gm(Sn) −→ {ωα : Mα(Sm,Sn)→ Sn; α ∈ πm(Sn)}/ fhe (1.2)
and there is a bijection
πm(Sn)/Gm(Sn)
∼=−→ {ωα : Mα(Sm, Sn)→ Sn; α ∈ πm(Sn)}/ sfhe. (1.3)
We point out that a generalization of the results above has been stated in [16]. As a consequence,
using the surjections (1.1) and (1.2), it is possible to obtain an upper bound for the number of
homotopy types of path-components for the mapping space M(Sn+k, Sn) and to the number of
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1026 M. GOLASIŃSKI, THIAGO DE MELO
evaluation fibrations ωα : Mα(Sn+k,Sn)→ Sn, for α ∈ πn+k(Sn), up to fibre-homotopy equivalence
(fhe), respectively. In addition, the bijection (1.3) gives the exactly number of evaluation fibrations,
up to strong fibre-homotopy equivalence (sfhe).
Remark 1.1. By [11] (Theorem 4.1) we have Mα(Sm,Sn) ' M0(Sm,Sn) if and only if
[ιn, α] = 0, if and only if α ∈ Gm(Sn). Thus if Gm(Sn) πm(Sn) then there are at least two
path-components which are not homotopy equivalent, that is, |πm(Sn)/±Gm(Sn)| ≥ 2, and there is
only one if and only if Gm(Sn) = πm(Sn).
We close this section with the following fact (on the relation ∼ defined above) useful in the
sequel. First, given reals x, y, write
χ(x, y) =
⌈
(x− 1)(y − 1)
2
⌉
+
⌈
x− 1
2
⌉
+
⌈
y − 1
2
⌉
+ 1,
where dre = min{k ∈ Z; k ≥ r} for any real r.
Lemma 1.1. For positive integers m,m′, n, n′ with m | n, m′ | n′ and n, n′ ≥ 1, let Zm ×
× Zm′ < Zn × Zn′ , mZ × Zm′ < Z × Zn′ and mZ × m′Z < Z × Z be the obvious inclusions.
Then
|(Zn × Zn′)/± (Zm × Zm′)| = χ
(
n
m
,
n′
m′
)
, (1.4)
|(Z× Zn′)/± (mZ× Zm′)| = χ
(
m,
n′
m′
)
, (1.5)
|(Z× Z)/± (mZ×m′Z)| = χ
(
m,m′
)
. (1.6)
In particular, |(Zn × Zn′)/± (Zm × Zn′)| = |Zn/± Zm| = χ
( n
m
, 1
)
.
Proof. For any (a, b) ∈ Zn × Zn′ , (a, b) ∼ (c, d) where 1 ≤ c ≤ n
m
, 1 ≤ d ≤ n′
m′
. Furthermore,
for c 6= n
m
and d 6= n′
m′
, (c, d) ∼
(
n
m
− c, n
′
m′
− d
)
for 1 ≤ d ≤ n′
m′
− 1 and then we have⌈
1
2
( n
m
− 1
)( n′
m′
− 1
)⌉
nonequivalent elements. In addition,
( n
m
, d
)
∼
(
n
m
,
n′
m′
− d
)
for 1 ≤
≤ d ≤ n′
m′
−1 and
(
c,
n′
m′
)
∼
(
n
m
− c, n
′
m′
)
for 1 ≤ c ≤ n
m
−1. So, we obtain more
⌈
1
2
( n
m
− 1
)⌉
+
+
⌈
1
2
(
n′
m′
− 1
)⌉
nonequivalent elements. Finally, since that the trivial element is
(
n
m
,
n′
m′
)
, the
equation (1.4) follows.
To prove (1.5) and (1.6), just replace
n
m
by m and
n
m
,
n′
m′
by m,m′ respectively.
Lemma 1.1 is proved.
2. Main results. We make use of [5] and Lemma 1.1 to estimate the cardinality
|πn+k(Sn)/±Gn+k(Sn)| (2.1)
for 8 ≤ k ≤ 13. We first recall the results from [6] for 0 ≤ k ≤ 7 and make some improvements of
the cardinality (2.1).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
EVALUATION FIBRATIONS AND PATH-COMPONENTS OF THE MAPPING SPACE M(Sn+k, Sn) . . . 1027
Proposition 2.1. The cardinality |πn+k(Sn)/±Gn+k(Sn)| for 0 ≤ k ≤ 7 is, respectively:
one, if n = 1, 3, 7; two, if n 6= 1, 3, 7 is odd; |Z| if n is even;
one, if n = 1, 2, 6 or n ≡ 3 (mod 4); two, otherwise;
one, if n = 1, 5 or n ≡ 2, 3 (mod 4); two, otherwise;
ten, if n = 4; one, if n ≡ 7 (mod 8) or n = 2i − 3 for i ≥ 3; two, if n ≡ 1, 3, 5 (mod 8) and
n ≥ 9 and n 6= 2i − 3; seven, if n ≡ 2 (mod 4) and n ≥ 6 or n = 12; thirteen, if n ≡ 0 (mod 4)
and n ≥ 8 and n 6= 12;
one, for all n ≥ 1;
one, if n 6= 6; two, otherwise;
one, if n ≡ 4, 5, 7 (mod 8) or n = 2i − 5 for i ≥ 4; two, otherwise;
one, if n = 5, 11 or n ≡ 15 (mod 16); two, if n is odd and n ≥ 9, unless n = 11 and n ≡ 15
(mod 16); eight, if n = 4; thirty one, if n = 6; ninety one, if n = 8; one hundred twenty one, if n is
even and n ≥ 10.
2.1. The case k = 8. Making use of the Gottlieb groups Gn+8(Sn) computed in [5] (Proposi-
tion 6.3) we estimate |πn+8(Sn)/±Gn+8(Sn)|.
For n = 1, 2, 6, 10 or n ≡ 3 (mod 4), the cardinality (2.1) is one.
For n ≡ 0, 1 (mod 4) and n 6= 8, 9, or n ≡ 22 (mod 32) and n ≥ 54, Gn+8(Sn) = 0 and then
(2.1) is equal to |πn+8(Sn)/ ± 0|, that is: two, if n = 4, 5, since that πn+8(Sn) = {εn} ∼= Z2; four,
if n ≥ 12, since that πn+8(Sn) = {ν̄n, εn} ∼= (Z2)2.
For n ≡ 2 (mod 8) and n ≥ 18, Gn+8(Sn) = {εn} ∼= Z2 and πn+8(Sn) = {ν̄n, εn} ∼= (Z2)2.
So the cardinality (2.1) is two. But [ιn, ν̄n] 6= 0 and then ων̄n is not fibre-homotopy equivalent to ω0
(which is fibre-homotopy equivalent to ωεn).
For n = 22, or n ≡ 14 (mod 16), or n ≡ 6 (mod 32) and n ≥ 14, Gn+8(Sn) = {ηnσn+1} ∼=
∼= Z2 and πn+8(Sn) = {ν̄n, εn} ∼= (Z2)2. Thus, the cardinality (2.1) is two. In view of [17] (Lemma
6.4), it holds ηnσn+1 = ν̄n + εn ∈ Gn+8(Sn) for n ≥ 9 and the bilinearity of the Whitehead product
yields [ιn, ν̄n] = −[ιn, εn]. By [10] (Theorem 2.3), ων̄n and ωεn are fibre-homotopy equivalent as
well as ω0 and ων̄n+εn .
For n = 8, the Gottlieb group is G16(S8) = {(Eσ′)η15, σ8η15 + ν̄8 + ε8} ∼= (Z2)2 and the
homotopy group is π16(S8) = {(Eσ′)η15, σ8η15, ν̄8, ε8} ∼= (Z2)4. We replace the generator σ8η15 ∈
∈ π16(S8) by the sum σ8η15 + ν̄8 + ε8 and then (2.1) is four.
For n = 9, G17(S9) = {[ι9, ι9]} ∼= Z2 and π17(S9) = {σ9η16, ν̄9, ε9} ∼= (Z2)3. Although the
generators for n = 9 are different from that ones for n = 8, but (2.1) is four as well.
We can summarize the results above and estimate the number of homotopy types of path-
components of the mapping space M(Sn+8,Sn) and fibre-homotopy equivalence types of evaluation
fibrations ωα : Mα(Sn+8,Sn)→ Sn for α ∈ πn+8(Sn).
Proposition 2.2. The cardinality |πn+8(Sn)/±Gn+8(Sn)| is:
one, if n = 1, 2, 6, 10 or n ≡ 3 (mod 4);
two, if n = 4, 5, 22, or n ≡ 2 (mod 8) and n ≥ 18, or n ≡ 14 (mod 16), or n ≡ 6 (mod 32)
and n ≥ 14;
four, if n ≡ 0, 1 (mod 4) and n ≥ 8, or n ≡ 22 (mod 32) and n ≥ 54.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1028 M. GOLASIŃSKI, THIAGO DE MELO
2.2. The case k = 9. In view of [5] (Proposition 6.4), we estimate the cardinality
|πn+9(Sn)/±Gn+9(Sn)|.
For n = 1, 2, 6 or n ≡ 3 (mod 4), |πn+9(Sn)/±Gn+9(Sn)| = 1.
For n ≡ 0 (mod 8) and n ≥ 16, Gn+9(Sn) = 0 and then (2.1) is
|πn+9(Sn)/± 0| = |(Z2)3/± 0| = 8.
For n ≡ 2 (mod 4) and n ≥ 14, or n = 2i − 7 with i ≥ 5, or n ≡ 5 (mod 8) and n 6≡ 53
(mod 64), Gn+9(Sn) ∼= (Z2)2 and πn+9(Sn) ∼= (Z2)3 and then (2.1) is two.
For n ≡ 4 (mod 8), or n ≡ 53 (mod 64) and n ≥ 117, or n ≡ 1 (mod 8) and n ≥ 17 and
n 6= 2i − 7, Gn+9(Sn) ∼= Z2 and πn+9(Sn) ∼= (Z2)3. So (2.1) is four.
For n = 8, G17(S8) = {(Eσ′)η2
15, σ8η
2
15 +ν3
8 +η8ε9} ∼= (Z2)2 and π17(S8) = {(Eσ′)η2
15, σ8η
2
15,
ν3
8 , µ8, η8ε9} ∼= (Z2)5. Replacing the generator σ8η
2
15 ∈ π17(S8) by the sum σ8η
2
15 + ν3
8 + η8ε9, (2.1)
is |{ν3
8 , µ8, η8ε9}/± 0| = |(Z2)3/± 0| = 8.
For n = 9, the Gottlieb group is G18(S9) = {σ9η
2
16, ν
3
9 , η9ε10} ∼= (Z2)3 and the homotopy group
is π18(S9) = {σ9η
2
16, ν
3
9 , µ9, η9ε10} ∼= (Z2)4. In a similar way we conclude that (2.1) is two.
Finally, for n = 10, G19(S10) = {3[ι10, ι10], ν3
10, η10ε11} ∼= 3Z ⊕ (Z2)2 and π19(S10) =
= {∆(ι21), ν3
10, µ10, η10ε11} ∼= Z ⊕ (Z2)3. So (2.1) is |(Z ⊕ (Z2)3)/ ± (3Z ⊕ (Z2)2)| = 4, by
Lemma 1.1.
Then, we summarize the results above as follows:
Proposition 2.3. The cardinality |πn+9(Sn)/±Gn+9(Sn)| is:
one, if n = 1, 2, 6, or n ≡ 3 (mod 4);
two, if n = 9, or n ≡ 2 (mod 4) and n ≥ 14, or n = 2i − 7 with i ≥ 5, or n ≡ 5 (mod 8) and
n 6≡ 53 (mod 64);
four, if n = 10, or n ≡ 4 (mod 8), or n ≡ 53 (mod 64) and n ≥ 117, or n ≡ 1 (mod 8) and
n ≥ 17 and n 6= 2i − 7;
eight, if n ≡ 0 (mod 8).
2.3. The cases k = 10,11. Following the same ideas as above and making use of Lemma 1.1,
we can also compute the appropriate quotient set to estimate its cardinality to state the next results:
Proposition 2.4. The cardinality |πn+10(Sn)/±Gn+10(Sn)| is:
one, if n = 1, 2, 5, or n ≡ 3 (mod 4);
two, if n ≡ 2 (mod 4), or n ≡ 1 (mod 4) and n ≥ 9;
four, if n ≡ 0 (mod 4).
Proposition 2.5. The cardinality |πn+11(Sn)/±Gn+11(Sn)| is:
one, if n ≡ 1 (mod 2) and n 6≡ 115 (mod 128);
two, if n ≡ 115 (mod 128) and n ≥ 243;
twenty two, two hundred fifty-four, seven hundred fifty seven, if n = 4, 8, 12 respectively;
two hundred fifty-three, if n ≡ 0 (mod 4) and n ≥ 16;
one hundred twenty-seven, if n ≡ 2 (mod 4) and n ≥ 6.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
EVALUATION FIBRATIONS AND PATH-COMPONENTS OF THE MAPPING SPACE M(Sn+k, Sn) . . . 1029
2.4. The cases k = 12,13. Following [5] (Section 6), we have Gn+12(Sn) = πn+12(Sn) for
n 6= 10 and Gn+13(Sn) = πn+13(Sn) for n = 2 or n odd. So the cardinality (2.1) is one. For
k = 12, n = 10 or k = 13, n even and n 6= 2, 4, 14, the cardinality (2.1) is two. For k = 13, n = 4,
the cardinality (2.1) is four and for k = 13, n = 14 it is five.
In resume:
Proposition 2.6. The cardinality |πn+k(Sn)/±Gn+k(Sn)| is:
one, for k = 12 and n 6= 10, or k = 13 and n = 2 or n odd;
two, for k = 12 and n = 10, or k = 13 and n even, n 6= 2, 4, 14;
four, for k = 13 and n = 4;
five, for k = 13 and n = 14.
Remark 2.1. We observe that the cases k = 9, n = 53 and k = 11, n = 115 are missing
because the Gottlieb groups G62(S53) and G126(S115) are unknown. On the other hand, the 2-primary
component of the homotopy group π126(S115) is π115
126 = {ζ115} [19] (Theorem 7.4) and in view of
[15] (Theorem 3.1) the Kervaire invariant θ6 exists in the stable homotopy group πs126 if and only if
[ζ115, ι115] = 0.
We recall that in [10] (Example 1), two fhe evaluation fibrations ωα : Mα(S2 ∨ S2,S2)→ S2 and
ωβ : Mβ(S2 ∨ S2,S2)→ S2 for α, β ∈ [S2 ∨ S2, S2] not being sfhe are constructed. From the results
above, we get:
Corollary 2.1. There are evaluation fibrations ωα : Mα(Sn+k, Sn)→ Sn for some α ∈ πn+k(Sn)
and 0 ≤ k ≤ 13 being fhe and not sfhe.
At the end of this section, we notice that:
Remark 2.2. The procedure above leads to an estimation of the number of homotopy types of
path-components of M(Sn+k,Sn)∗ and fibre-homotopy types of evaluation fibrations ωα : Mα(Sn+k,
Sn)∗ → Sn with 0 ≤ k ≤ 13.
3. Applications to projective spaces.. Let R and C be the fields of real and complex numbers,
respectively and H the skew R-algebra of quaternions. In this section we apply the results above to
study the path-components of M(Sm,FPn) for F = R,C,H and M(Sm,KP 2), where K denotes
the Cayley algebra.
Denote by FPn the n-projective space over F. Put d = dimR F, write im,n : FPm ↪→ FPn,
m ≤ n, for the inclusion map, γn = γn,F : S(n+1)d−1 → FPn for the quotient map and set iF =
= i1,n : FP 1 = Sd ↪→ FPn. Let EX be the suspension of a space X and denote by E : πm(X) →
→ πm+1(EX) the suspension homomorphism. Next, write ∆ = ∆FP : πm(FPn) → πm−1(Sd−1)
for the connecting map. By [3] (Theorem (2.1)) it holds:
∆(iF∗E) = idπm−1(Sd−1)
and
πm(FPn) = γn∗πm(Sd(n+1)−1)⊕ iF∗Eπm−1(Sd−1).
Hence, πm(RP 1) ∼= πm(S1) and πm(CP 1) ∼= πm(S2) for m ≥ 0. Further, for n > 1, we derive
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1030 M. GOLASIŃSKI, THIAGO DE MELO
πm(RPn) =
0, if m = 0,
Z2, if m = 1,
γn∗πm(Sn), if m > 1,
and
πm(CPn) =
0, if m = 0, 1,
Z, if m = 2,
γn∗πm(S2n+1), if m > 2.
The path-connected components of M(Sm,FPn) are in one-to-one correspondence with the
set [Sm,FPn] of (free) homotopy classes. Because CPn and HPn are 1-connected, [Sm,RPn] ∼=
∼= πm(RPn)/π1(RPn) and [Sm,CPn] ∼= πm(CPn), [Sm,HPn] ∼= πm(HPn).
By [2] (Corollary (7.4)) and [3] ((4.1) – (4.3)), we obtain a formula:
Lemma 3.1. Let h0α ∈ πm(S2n−1) be the 0-th Hopf – Hilton invariant for α ∈ πm(Sn). Then
[γnα, iR] =
0 for odd n;
(−1)mγn(−2α+ [ιn, ιn] ◦ h0α) for even n.
Let τη(ξ) ∈ πm(X) be the operation of η ∈ π1(X) on ξ ∈ πm(X). Then, in view of [21]
(Chapter X, (7.6)), it holds
[ξ, η] = (−1)m(τη(ξ)− ξ).
Hence, by Lemma 3.1, the action of π1(RPn) on πm(RPn) is trivial for odd n and we get
[Sm,RPn] ∼= πm(RPn) = γn∗πm(Sn). Further, the map γn : S(n+1)d−1 → FPn leads to com-
mutative diagrams of surjective maps
πm(Sn)/±Gm(Sn) −→ {Mα(Sm,Sn); α ∈ πm(Sn)}/ '
↓ ↓
πm(RPn)/± γn∗Gm(Sn) −→ {Mα(Sm,RPn); α ∈ πm(RPn)}/π1(RPn)/ '
and
πm(S2n+1)/±Gm(S2n+1) −→ {Mα(Sm, S2n+1); α ∈ πm(S2n+1)}/ '
↓ ↓
πm(CPn)/± γn∗Gm(S2n+1) −→ {Mα(Sm,CPn); α ∈ πm(CPn)}/ ' .
Further, πm(HPn) = γn∗πm(S4n+3) ⊕ iH∗Eπm−1(S3). Because Gm(S3) = πm(S3), the path-
components Mα(Sm,HPn) for α ∈ iH∗Eπm−1(S3) have the same homotopy type. This yields the
next commutative diagram of surjective maps
πm(S4n+3)/±Gm(S4n+3) −→ {Mα(Sm,S4n+3); α ∈ πm(S4n+3)}/ '
↓ ↓
πm(HPn)/± γn∗Gm(S4n+3) −→ {Mα(Sm,HPn); α ∈ πm(HPn)}/ ' .
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
EVALUATION FIBRATIONS AND PATH-COMPONENTS OF THE MAPPING SPACE M(Sn+k, Sn) . . . 1031
Consequently, the main result presented in Section 2 leads to estimations of |{Mα(S(n+1)d−1+k,
FPn)}/ ' | for k ≤ 13 and F = R,C,H. Then, the results [9] (Theorems 1, 2) and [10] (Theo-
rem 2.3) lead also to:
Remark 3.1. There are estimations of fibre-homotopy types of evaluation fibrations ωα:
Mα(S(n+1)d−1+k,FPn) → FPn and their strong fibre-homotpy types for k ≤ 13 and F = R,C,H
as well.
Next, write KP 2 = S8 ∪σ8 e16 for the Cayley projective plane and iK : S8 ↪→ KP 2 for the
inclusion map, where σ8 : S15 → S8 is the Hopf map. Then, in view of [17], it holds πm(KP 2) =
= iK∗Eπn−1(S7) ∼= πm−1(S7) for m ≤ 21. Because Gm(S7) = πm(S7), all path-components of
M(Sm,KP 2) have the same homotopy type for m ≤ 21.
4. Miscellanea on mapping spaces. Homotopy properties of various path-components
Mα(Sm, Sn) have been studied in [1, 14, 20] and then some homotopy groups πk(Mα(Sm,Sn))
computed. However, the rational type of M(Sm, Sn) and M(Sm,Sn)∗ has been fully described in
[4, 18] as follows:
Theorem 4.1. (i) For n odd and any m:
M(Sm,Sn) ∼=Q
Sn ×K(Z, n−m), if n > m,∐∞
k=1
Sn, if n = m,
Sn, if n < m,
M(Sm,Sn)∗ ∼=Q
K(Z, n−m), if n > m,∐∞
k=1
∗, if n = m,
∗, if n < m.
(ii) For n even and any m:
M(Sm,Sn) ∼=Q
Y, if n > m,
Sn ×K(Z, 2n−m− 1)
∐∞
k=1
S2n−1, if n = m,
Sn ×K(Z, 2n−m− 1), if n < m < 2n− 1,∐∞
k=1
Sn, if m = 2n− 1,
Sn, if m > 2n− 1,
where Y is given by the fibration Sn ×K(Z, n−m)→ Y → K(Z, 2n−m− 1);
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1032 M. GOLASIŃSKI, THIAGO DE MELO
M(Sm, Sn)∗ ∼=Q
K(Z, n−m)×K(Z, 2n−m− 1), if n > m,∐∞
k=1
K(Z, 2n−m− 1), if n = m,
K(Z, 2n−m− 1), if n < m < 2n− 1,∐∞
k=1
∗, if m = 2n− 1,
∗, if m > 2n− 1.
Now, let A be an abelian group and n ≥ 1. A spaceM(A, n) such that
H̃i(M(A, n)) =
A, if i = n,
0, otherwise
is called a Moore space of type (A, n). If A = Zk is a cyclic group of order k then such space can
be constructed from the n-sphere Sn by attaching an (n + 1)-cell en+1 via a map f : Sn → Sn of
degree k.
Proposition 4.1 ([12], Proposition 4H.2). For any n > 1, and any abelian group A and a
pointed space X there are natural short exact sequences
0→ Ext(A, πn+1(X))→ [M(A, n), X]∗ → Hom(A, πn(X))→ 0. (4.1)
Notice that for A = Zk, we get
Ext(Zk, πn+1(X)) ∼= Zk ⊗ πn+1(X) ∼= πn+1(X)/kπn+1(X)
and
Hom(Zk, πn(X)) = kπn(X) = {α ∈ πn(X); kα = 0}.
Hence, the sequence (4.1) leads to
0→ πn+1(X)/kπn+1(X)→ [M(Zk, n), X]∗ → kπn(X)→ 0,
which we use to compute [M(Zk, n), Sm]∗ (in fact [M(Zk, n),Sm]) for some m,n.
The case m = 1 is simple: if n = 1 then kπn(S1) = 0 and πn+1(S1) = πn(S1) = 0 for n > 1.
Thus, we have [M(Zk, n),S1]∗ = [M(Zk, n),S1] = 0.
From now on, we assume that m > 1. So, π1(Sm) = 0 and [M(Zk, n),Sm]∗ = [M(Zk, n), Sm].
Case 1. If n+ 1 < m, then πn(Sm) = πn+1(Sm) = 0. So, [M(Zk, n),Sm] = 0.
Case 2. If n+ 1 = m, then πn+1(Sm) ∼= Z and πn(Sm) = 0 which imply that [M(Zk, n), Sm] ∼=
∼= Zk.
Case 3. If n + 1 > m, then n = m + l − 1, for some l > 0. Now we study the short exact
sequences below for l > 0
0→ πm+l(Sm)/kπm+l(Sm)→ [M(Zk,m+ l − 1),Sm]→ kπm+l−1(Sm)→ 0. (4.2)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
EVALUATION FIBRATIONS AND PATH-COMPONENTS OF THE MAPPING SPACE M(Sn+k, Sn) . . . 1033
First, if l = 1, then kπm+l−1(Sm) = 0 and we have to consider the cases m = 2 and m > 2
separately, since π3(S2) ∼= Z and πm+1(Sm) ∼= Z2, respectively. More precisely,
[M(Zk,m), Sm] ∼= πm+1(Sm)/kπm+1(Sm) ∼=
Zk, if m = 2,
Z2, if m > 2 and k is even,
0, if m > 2 and k is odd.
Next, if l = 2, then πm+l(Sm) ∼= Z2 and πm+l−1(Sm) ∼= Z for m = 2 and πm+l−1(Sm) ∼= Z2
for m > 2. If m = 2, then the sequence (4.2) yields
[M(Zk, 3), S2] ∼= Z2/kZ2
∼=
Z2, if k is even,
0, if k is odd.
If m > 2, then (4.2) becomes 0 → Z2/kZ2 → [M(Zk,m + 1), Sm] → kZ2 → 0 and if k is odd,
then [M(Zk,m+ 1), Sm] = 0, while if k is even, then 0→ Z2 → [M(Zk,m+ 1),Sm]→ Z2 → 0.
So, we get |[M(Zk,m+ 1),Sm]| = 4.
Further, if l = 3, then
πm+3(Sm) ∼=
Z2, if m = 2,
Z12, if m = 3,
Z⊕ Z12, if m = 4,
Z24, if m ≥ 5,
and πm+2(Sm) ∼= Z2. Since kZ2 = 0 for any odd k, we obtain
[M(Zk,m+ 2), Sm] ∼=
0, if m = 2,
Z4, if m = 3 and 3 | k,
0, if m = 3 and 3 - k,
(Z⊕ Z12)/k(Z⊕ Z12), if m = 4,
Z24/kZ24, if m ≥ 5.
If k is even, then kZ2 = Z2 and in view of (4.2) we get
0→ πm+3(Sm)/kπm+3(Sm)→ [M(Zk,m+ 2),Sm]→ Z2 → 0
which leads to the value of |[M(Zk,m + 2), Sm]|. Following the procedure above and using the
homotopy groups πm+l(Sm) (see, e.g., [19]), it is possible to determine |[M(Zk,m + l), Sm]| for
other values of l > 3 as well.
Acknowledgements. This paper was started during the visit of the second author to the Faculty
of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń (Poland), December
2010–March 2011. He would like to thank that Faculty for its hospitality during his stay. This visit
was supported by PROPG, UNESP–Univ Estadual Paulista (Brazil).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1034 M. GOLASIŃSKI, THIAGO DE MELO
1. Abe M. Über die stetigen Abbildungen der n-Sphäre in einen metrischen Raum // Jap. J. Math. – 1940. – 16. –
S. 169 – 176.
2. Barcus W. D., Barratt M. G. On the homotopy classification of the extensions of a fixed map // Trans. Amer. Math.
Soc. – 1958. – 88. – P. 57 – 74.
3. Barratt M. G., James I. M., Stein N. Whitehead products and projective spaces // J. Math. and Mech. – 1960. – 9. –
P. 813 – 819.
4. Buijs U., Murillo A. Basic constructions in rational homotopy of function spaces // Ann. Inst. Fourier (Grenoble). –
2006. – 56, № 3. – P. 815 – 838.
5. Golasiński M., Mukai J. Gottlieb groups of spheres // Topology. – 2008. – 47, № 6. – P. 399 – 430.
6. Golasiński M. Evaluation fibrations and path-components of the map space M(Sn+k, Sn) for 0 ≤ k ≤ 7 // Mat.
Stud. – 2009. – 31, № 2. – P. 189 – 194.
7. Gonçalves D., Koschorke U., Libardi A., Neto O. M. Coincidences of fiberwise maps between sphere bundles over
the circle // Proc. Edinburgh Math. Soc. (to appear).
8. Gottlieb D. A certain subgroup of the fundamental group // Amer. J. Math. – 1965. – 87. – P. 840 – 856.
9. Gottlieb D. Evaluation subgroups of homotopy groups // Amer. J. Math. – 1969. – 91. – P. 729 – 756.
10. Hansen V. L. Equivalence of evaluation fibrations // Invent. math. – 1974. – 23. – P. 163 – 171.
11. Hansen V. L. The homotopy problem for the components in the space of maps of the n-sphere // Quart. J. Math. –
1974. – 25. – P. 313 – 321.
12. Hatcher A. Algebraic topology. – Cambridge: Cambridge Univ. Press, 2002.
13. Hu S. T. Homotopy theory // Pure and Appl. Math. – New York; London: Acad. Press, 1959. – Vol. 8.
14. Koh S. S. Note on the homotopy properties of the components of the mapping space XSp
// Proc. Amer. Math. Soc. –
1960. – 11. – P. 896 – 904.
15. Lam K. Y., Randall D. Block bundle obstruction to Kervaire invariant one // Contemp. Math. – 2006. – 407. –
P. 163 – 171.
16. Lupton G., Smith S. B. Criteria for components of a function space to be homotopy equivalent // Math. Proc.
Cambridge Phil. Soc. – 2008. – 145, № 1. – P. 95 – 106.
17. Mimura M. The homotopy groups of Lie groups of low rank // J. Math. Kyoto Univ. – 1967. – 6. – P. 131 – 176.
18. Murillo A. Rational homotopy type of free and pointed mapping spaces between spheres // Brazilian-Polish Topology
Workshop. – Toruń, Warsaw, 2012.
19. Toda H. Composition methods in homotopy groups of spheres // Ann. Math. Stud. – Princeton, N.J.: Princeton Univ.
Press, 1962. – 49.
20. Whitehead G. W. On products in homotopy groups // Ann. Math. – 1946. – 47, № 2. – P. 460 – 475.
21. Whitehead G. W. Elements of homotopy theory // Grad. Texts Math. – 1978. – 61.
Received 05.08.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
|