Certain subgroups of groups of self-pair homotopy equivalences

Topology and its Applications 264 (2019) 382–393

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Topology and its Applications www.elsevier.com/locate/topol

Certain subgroups of groups of self-pair homotopy equivalences Hye Seon Shin a , Kee Young Lee b,2 , Ho Won Choi c,∗,1 a b c

Department of Mathematics, Korea University, Seoul 702-701, Republic of Korea Division of Applied Mathematical Sciences, Korea University, Sejong City, 30019, Republic of Korea Institute of Natural Science, Korea University, Sejong 339-700, Republic of Korea

a r t i c l e

i n f o

Article history: Received 16 May 2018 Accepted 31 October 2018 Available online 5 June 2019 MSC: 55P10 55P30 54E30

a b s t r a c t Let E(X) be the set of based homotopy classes of based self-homotopy equivalences of a CW-complex X. The concept of E(X) is applied to the category of pairs and is extended to a general concept E(α) for a map α : A → B. In this study, E(α) is generalized to Eγ (α) for two objects α and γ. Several generalized subgroups of E(α) or Eγ (α) are obtained and are combined to form an exact sequence. The exactness and the split property of this sequence is investigated. In particular, the sequence of a product space or a wedge space is demonstrated to be a split exact sequence. The split property and the exactness are used to completely compute those subgroups. © 2019 Elsevier B.V. All rights reserved.

Keywords: Category of pairs Self pair of homotopy equivalences

1. Introduction Let X be a connected CW complex with base point ∗, and let E(X) be the set of based homotopy classes of based self-homotopy equivalences. Then, the composition of homotopy classes induces a group structure on E(X). E(X) and certain natural subgroups thereof are fundamental objects in homotopy theory and have been extensively studied. In particular, the concept of E(X) was applied to the category of pairs and was extended to a general concept E(α) for a map α : A → B [9]. Moreover, En (X), which comprises all homotopy equivalences inducing identity map on homotopy groups up to n dimension, was introduced and studied in [5]. For a survey of results and applications of E(X), the reader is referred to [1] and [10], and for computing subgroups and theories of E(X), to [1], [3], [4], [5], [6] and [7]. * Corresponding author. E-mail addresses: [email protected] (H.S. Shin), [email protected] (K.Y. Lee), [email protected] (H.W. Choi). The corresponding author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1C1A1A01055455). 2 The second-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07045599). 1

https://doi.org/10.1016/j.topol.2019.06.012 0166-8641/© 2019 Elsevier B.V. All rights reserved.

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In this study, a more general concept Eγ (α) than E(α) is introduced for two objects α and γ in the category of pairs, and it is a subgroup of E(α) as well as a homotopy invariant. Moreover, the special subgroup E(i) for the inclusion map i : A → X is extensively studied, as well as its subgroups En (i) and En (i; idA ). These are the subgroups of elements in E(i) that induce the identity map on the homotopy groups up to n. They are combined to obtain a sequence 1

En (i; idA )

inc.

πA

En (i)

En (A),

where inc. : E(i; idA ) → E(i) is the inclusion and πA is the projection onto the first coordinate. If A is a retract of X, then the sequence is exact. Furthermore, if i is the inclusion from X to X × Y or X ∨ Y , then the sequence is a split exact sequence. The exactness or the split property are used to show the following facts: Let X and Y be the m-dimensional sphere S m and n-dimensional sphere S n , respectively, where m ≥ n ≥ 1. For the inclusions iY : Y → X × Y and iX : X → X × Y , and for m = 3, 7 and n = 1, it holds that

Ek (iX )

k ∼ = E (iY )

k ≥ m, 1

m>k≥1

1

Z2

Let M1 = M (Z2 , n + 1) and M2 = M (Z3 , n) be Moore spaces for n ≥ 5 and X = M1 ∨ M2 . Then, Edim X (X) ∼ = Edim X (iM1 ) ∼ = Edim X (iM2 ) ∼ = Z2 , whereas Edim X (iM1 ; idM1 ) is not isomorphic to Edim X (iM2 ; idM2 ). Henceforth, it will be assumed that all topological spaces are based connected CW complexes and all maps and homotopies are base point preserving. The set of based homotopy classes of based maps from X to Y is denoted by [X, Y ]. Notationally, a map f : X → Y and its homotopy class [f ] in [X, Y ] will not be distinguished. 2. Preliminaries In this section, the category of pairs is first reviewed. It is the category whose objects are maps between two based spaces and a morphism from one object α : (X1 , ∗) → (X2 , ∗) to another object β : (Y1 , ∗) → (Y2 , ∗) is a pair of maps (f1 , f2 ) such that the diagram X1

α

f1

Y1

X2 f2

β

Y2

is commutative, i.e., f2 ◦ α = β ◦ f1 . The pair maps (f1 , f2 ) and (g1 , g2 ) are pair homotopic if the following diagram commutes: X1 × I

α×I

F1

Y1

X2 × I F2

β

Y2

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where F1 and F2 are homotopies f1  g1 and f2  g2 , respectively. The homotopy class of (f1 , f2 ) is denoted by [f1 , f2 ]. (f1 , f2 ) is called a homotopy equivalent morphism or simply a homotopy equivalence, if there is a morphism (g1 , g2 ) such that (g1 , g2 ) ◦ (f1 , f2 )  (idX1 , idX2 ) and (f1 , f2 ) ◦ (g1 , g2 )  (idY1 , idY2 ). In this case, (g1 , g2 ) is called a homotopy inverse of (f1 , f2 ). Furthermore, (f1 , f2 ) is called a self homotopy equivalent morphism or simply a self homotopy equivalence, if α = β, and it is called a self pair homotopy equivalent morphism or simply, a self-pair homotopy equivalence, if α = β = i : A → X is the inclusion. Definition 2.1. ([9]) For an object α : X1 → X2 , let E(α) = {[f1 , f2 ]|(f1 , f2 ) is a homotopy equivalence in Π(α, α)}. Certain preliminary results are now presented. Proposition 2.2. ([2, Theorem 3.8]) For Moore space X = M (G, n), (1) Edim (X) ∼ = E∗ (X) ∼ = ⊕(r+s)r Z2 dim +1 ∼ (2) E (X) = 1 for n > 3, where r is the rank of G and s is the number of 2-torsion summands in G. Proposition 2.3. ([10, Theorem 2.1])  E(M (Zq , n)) ∼ =

Zq  Z∗q

Z(2,q) ×

n = 2, Z∗q

n ≥ 3,

where, for the semi-direct product, the automorphism group Z∗q of Zq acts in the usual manner. Let M1 = M (Zq , n + 1) and M2 = M (Zp , n), where p and q are positive integers. Lemma 2.4. ([5, Lemma 3.2]) Let d be the greatest common divisor of p and q. Then,  Zd {π2∗ (i1 )} [M2 , M1 ] ∼ = 0

if d = 1, if d = 1.

Lemma 2.5. ([5, Lemma 3.3]) If p or q is odd, then [M1 , M2 ] ∼ = 0. 3. Certain subgroups related to self-pair homotopy equivalent morphisms In this section, certain subgroups of E(α) are considered. Definition 3.1. For two objects α : X1 → X2 and γ : Y1 → Y2 , let Eγ (α) = {[f1 , f2 ] ∈ E(α)|(f1 , f2 ) is the identity from Π(γ, α) to itself}, where Π(γ, α) is the set of homotopy classes of maps from γ to α. If γ = α, then Eα (α) = {[f1 , f2 ] ∈ E(α)|(f1 , f2 ) is the identity from Π(α, α) to itself}.

(1)

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Furthermore, if γ : ∗ → ∗, then Eγ (α) = E(α). Thus, Eγ (α) is a generalization of E(α). Proposition 3.2. Eγ (α) is a subgroup of E(α). Proof. For given elements [f1 , f2 ], [f1 , f2 ] ∈ Eγ (α), we have [f1 , f2 ] ◦ [f1 , f2 ] ∈ Eγ (α) because ([f1 , f2 ] ◦ [f1 , f2 ]) = [f1 , f2 ] ◦ [f1 , f2 )] = idΠ(γ,α) ◦ idΠ(γ,α) = idΠ(γ,α) . For [idA , idX ] ∈ E(α), we have (idA , idX ) [g1 , g2 ] = [idA ◦ g1 , idX ◦ g2 ] = [g1 , g2 ] for any [g1 , g2 ] ∈ Π(γ, α). Thus, [idA , idX ] ∈ Eγ (α). For each [f1 , f2 ] ∈ Eγ (α), let [h1 , h2 ] be the inverse element of [f1 , f2 ] in E(α). As the diagram Y1

γ

g1

X1

g2 α

h1

X1

Y2

X2 h2

α

X2

is commutative for any [g1 , g2 ] ∈ Π(γ, α), we have (h1 , h2 ) [g1 , g2 ] = [h1 ◦ g1 , h2 ◦ g2 ] = idΠ(γ,α) ([h1 ◦ g1 , h2 , ◦g2 ]) = (f1 , f2 ) [h1 ◦ g1 , h2 ◦ g2 ] = [f1 ◦ h1 ◦ g1 , f2 ◦ h2 ◦ g2 ] = [g1 , g2 ]. Therefore, [h1 , h2 ] ∈ Eγ (α).

2

Proposition 3.3. If α and β have the same homotopy type, then Eγ (α) and Eγ (β) are isomorphic for any object γ. Proof. Suppose that α : X1 → X2 and β : Y1 → Y2 have the same homotopy type by a homotopy equivalent morphism (e1 , e2 ) : α → β with the homotopy inverse morphism (e1 , e2 ) : β → α. Let Φ : Eγ (α) → Eγ (β) be defined by Φ([f1 , f2 ]) = [(e1 , e2 ) ◦ (f1 , f2 ) ◦ (e1 , e2 )]. Since [f1 , f2 ] = [idπ(γ,α) ], we have [(e1 , e2 ) ◦ (f1 , f2 ) ◦ (e1 , e2 )] = [e1 , e2 ] ◦ [f1 , f2 ] ◦ [e1 , e2 ] = [e1 , e2 ] ◦ [e1 , e2 ] = [e1 ◦ e1 , e2 ◦ e2 ] = idΠ(γ,β) and

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Φ([f1 , f2 ] ◦ [g1 , g2 ]) = [e1 ◦ f1 ◦ g1 ◦ e1 , e2 ◦ f2 ◦ g2 ◦ e2 ] = [e1 ◦ f1 ◦ e1 ◦ e1 ◦ g1 ◦ e1 , e2 ◦ f2 ◦ e2 ◦ e2 ◦ g2 ◦ e2 ] = Φ([f1 , f2 ]) ◦ Φ([g1 , g2 ]). Thus, Φ is well-defined and a homomorphism. Conversely, let a homomorphism Θ : Eγ (β) → Eγ (α) be defined by Θ([h1 , h2 ]) = [e1 ◦ h1 ◦ e1 , e2 ◦ h2 ◦ e2 ]. Then, Θ is an inverse homomorphism of Φ. Therefore, Φ is an isomorphism. 2 Let Γ = {γ1 , γ2 , . . . , γn } be a set of objects in the category of pairs. A subgroup of E(α) will now be defined depending on this set that is also a subgroup of Eγi (α) for i = 1, 2, . . . , n. Definition 3.4. Let EΓ (α) = {[f1 , f2 ] ∈ E(α)|(f1 , f2 ) = idΠ(βi ,α) : Π(βi , α) → Π(βi , α) for i = 1, 2, . . . , n}. Equivalently, EΓ (α) = ∩ni=1 Eγi (α). Remark 1. (1) Let α : ∗ → X2 be a constant map and γi : ∗ → S i be a constant map to the i-dimensional sphere. Then, EΓ (α) = En (X2 ). Indeed, Eγi (α) = {[∗, f ] ∈ E(α) = E(X2 )|[∗, f ] = idΠ(γi ,α) } and Π(γi , α) ∼ = [S i , X2 ] = πi (X). Identifying [∗, f ] with [f ], we have EΓ (α) = {[f ] ∈ E(X)|[f ] = idπi (X2 ) for all 1 ≤ i ≤ n} = En (X2 ). (2) Let α : X1 → ∗ and γi : S i → ∗. Then, EΓ (α) = En (X1 ). (3) Let α : X1 → X2 and γi : ∗ → S i . Then EΓ (α) is denoted by En (α). Another subgroups of E(α) are now introduced. Definition 3.5. For any object α : X1 → X2 , subsets E(α; idX1 ) and En (α; idX1 ) of E(α) are defined by E(α; idX1 ) = {[idX1 , f ] ∈ E(α)|[idX1 ] is the identity of [X1 , X1 ]} and En (α; idX1 ) = E(α; idX1 ) ∩ En (α), respectively. These subsets are also subgroups of E(α).

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4. Certain exact sequences related to self pair homotopy equivalences As mentioned above, if En (i) and En (i; idA ) are the subgroup of E(i) for the inclusion map i : A → X, a sequence can be formed as follows. Theorem 4.1. If r : X → A is a retraction, then there is an exact sequence: 1

En (i; idA )

inc.

En (i)

πA

En (A),

where inc. : E(i; idA ) → E(i) is the inclusion and πA is the projection onto the first coordinate. k = idπk (X) and πA ([f1 , f2 ]) = [f1 ]. Since r : X → A is a retraction, Proof. For [f1 , f2 ] ∈ En (i), we have f2 it holds that k f1 = (r ◦ i ◦ f1 )k

= (r ◦ f2 ◦ i)k k = rk ◦ f2 ◦ ik

= rk ◦ ik = (r ◦ i)k = idkA for k ≤ n. Thus, πA ([f1 , f2 ]) = [f1 ] ∈ En (A). If [f1 , f2 ] ∈ ker(πA ), then πA ([f1 , f2 ]) = [f1 ] = [idA ]. For all [g1 , g2 ] ∈ k E (i; idA ), g1  idA . Therefore, ker(πA ) is the image of inc. 2 A sufficient condition is now provided whereby the exact sequence in Theorem 4.1 can be a short exact sequence. Theorem 4.2. Let X ×Y be the product of two spaces. Then, we have the following split short exact sequence: En (i; idY )

1

inc.

En (i)

πY s

En (Y )

1,

where i : Y → X × Y and inc. : E(i; idY ) → E(i) are the corresponding inclusions. Proof. Since i : Y → X × Y is the inclusion, there exists a retraction r : X × Y → Y . Thus, by Theorem 4.1, we have the following exact sequence: 1

En (i; idY )

inc.

En (i)

πY

En (Y ).

Let s : En (Y ) → En (i) be the map defined by s([f ]) = [f, idX × f ]. Then, (idX × f )k = idkX × fk = idπk (X) × idπk (Y ) for k ≤ n. Thus, s is well-defined. Since πY ◦s([f ]) = πY ([f, idX ×f ]) = [f ], πY is a surjective homomorphism and the sequence is split. That is, we have the following split short exact sequence:

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388

En (i; idY )

1

inc.

En (i)

πY

En (Y )

s

1.

2

Proposition 4.3. Let X and Y be CW complex which are homotopy associative and inversive H-spaces. It is assumed that the sets [X ∧ Y, X × Y ] and [X, Y ] are trivial. Then, En (i, idY ) is isomorphic to En (X), where i : Y → X × Y is the inclusion. Moreover, there exists a split short exact sequence 1

En (X)

En (i)

En (Y )

1.

Proof. By Theorem 4.2, there exists a split short exact sequence 1

En (i; idY )

En (i)

En (Y )

1.

Let [g] ∈ En (X), and let a map γ : En (X) → En (i; idY ) be defined by γ([g]) = [idY , g × idY ]. For all n, πn (X × Y ) is isomorphic to πn (X) ⊕ πn (Y ). For any [δ] ∈ πn (X × Y ), [δ] can be identified with [pX ◦ δ, pY ◦ δ], where pX : X × Y → X and pY : X × Y → Y are projection maps. For k ≤ n, we have (g × idY )k [δ] = (g × idY )k [pX ◦ δ, pY ◦ δ] = [g ◦ pX ◦ δ, idY ◦ pY ◦ δ] = [pX ◦ δ, pY ◦ δ] = [δ]. Thus, [g × idY ] induces the identity map on πk (X × Y ) for k ≤ n. Therefore γ is well-defined. Conversely, let [idY , h] ∈ En (i; idY ). Since Y is a retract of X × Y , we have the following commutative diagram: 0

πk (Y )

j

idY 

0

πk (Y )

πk (X × Y ) h

j

πk (X × Y )

πk (X × Y, Y )

0

h

πk (X × Y, Y )

0

for any k ≥ 0, where idY  and the second h are isomorphisms. By the “five lemma”, the first h is an isomorphism. Furthermore, h induces the identity map on πk (X × Y ) for k ≤ n. Hence, [h] ∈ En (X × Y ). By [9, Theorem 4.4], [idY , h] can be identified with [idY , (pX ◦ h ◦ iX ) × idY )], where iX : X → X × Y is an inclusion map and hk = idπk (X×Y ) = idπk (X) × idπk (Y ) for k ≤ n. We have hk = ((pX ◦ h ◦ iX ) × idY ))k = (pX ◦ h ◦ iX )k × idY k = idπk (X) × idπk (Y ) . Thus, [pX ◦ h ◦ iX ] ∈ En (X).

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Let ξ : Ek (i; idY ) → Ek (X) be defined by ξ([idY , h]) = [pX ◦ h ◦ iX ]. Then, ξ is well-defined. (γ ◦ ξ)(idY , h) = γ(pX ◦ h ◦ iX ) = (idY , (pX ◦ h ◦ iX ) × idY ) = (idY , h) and (ξ ◦ γ)(g)(x) = ξ(idY , g × idY )(x) = pX ◦ (g × h ◦ iX ) ◦ iX (x) = pX ◦ (g(x) × h ◦ iX (∗)) = g(x). Therefore, En (i; idY ) is isomorphic to En (X). Consequently, we have the following split short exact sequence: 1

En (X)

En (i)

En (Y )

1.

2

Theorem 4.4. Let X and Y be (m − 1)-connected and ( − 1)-connected, respectively. Then, we have a split short exact sequence: En (i; idY )

1

En (i)

En (Y )

1,

where i : Y → X ∨ Y is an inclusion and n ≤ m +  − 1. Proof. By [2, Proposition 2.4], we have πk (X ∨ Y ) = πk (X) ⊕ πk (Y ) for n ≤ m +  − 1. 2 In [8], it was shown that if X and Y are CW homotopy associative and inversive co-H-spaces and the sets [X ∧ Y, X Y ] and [X, Y ] are trivial, where X Y = Σ(ΩX ∧ ΩY ), then there exists a split exact sequence. Corollary 4.5. Let X and Y be homotopy associative and inversive co-H-spaces such that the sets [X∧Y, X Y ] and [X, Y ] are trivial. If X and Y are (m −1)-connected and ( −1)-connected, respectively, then there exists split exact sequence 1

En (X)

En (i)

En (Y )

1,

where i : Y → X ∨ Y is an inclusion and n ≤ m +  − 1. Proof. By [8], E(X) is isomorphic to E(i; idY ). Thus, the proof is similar to that of Proposition 4.3.

2

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5. Examples of self-homotopy equivalent morphisms inducing the identity Example 1. Let X be the m-dimensional sphere S m and Y be the n-dimensional sphere S n , where m ≥ n ≥ 1. For the inclusions iY : Y → X ×Y and iX : X → X ×Y , by Theorem 4.2, we have two split exact sequences: 1

Ek (iX ; idS m )

Ek (iX )

Ek (S m )

1,

Ek (iY ; idS n )

Ek (iY )

Ek (S n )

1.

1 Then,

Ek (iX ) ∼ = Ek (iX ; idS m ) ⊕ Ek (S m ) and Ek (iY ) ∼ = Ek (iY ; idS n ) ⊕ Ek (S n ). Case 1. Let k ≥ m, n ≥ 1. As Ek (S m ) = 1 and Ek (S n ) = 1, Ek (iX ) ∼ = Ek (iX ; idS m ), Ek (iY ) ∼ = Ek (iY ; idS n ). Case 2. Let m > k ≥ n ≥ 1. As Ek (S m ) ∼ = Z2 and Ek (S n ) = 1, Ek (iX ) ∼ = Ek (iX ; idS m ) ⊕ Z2 , Ek (iY ) ∼ = Ek (iY ; idS n ). Case 3. Let m, n > k ≥ 1. As Ek (S m ) ∼ = Z2 and Ek (S n ) ∼ = Z2 , Ek (iX ) ∼ = Ek (iX ; idS m ) ⊕ Z2 , Ek (iY ) ∼ = Ek (iY ; idS n ) ⊕ Z2 . Case 4. Let m = 3, 7 and n = 1. Then, S m and S 1 are H-spaces. It holds that [S m ∧ S 1 , S 1 ] = 0, [S , S 1 ] = 0, and [S 1 , S m ] = 1. By Proposition 4.3, we have m

Ek (iX ) ∼ = Ek (S 1 ) ⊕ Ek (S m ) and Ek (iY ) ∼ = Ek (S m ) ⊕ Ek (S 1 ). Therefore, Ek (iX ) ∼ = Ek (iY ) but Ek (iX , idS m )  Ek (iY , idS 1 ). By Case 1 and 2, we have the following table: k ≥ m, n

m>k≥n

m=n=k

m, n > k

Ek (iX )

Ek (iX ; idS m )

Ek (iX ; idS m ) ⊕ Z2

Ek (iX ; idS m )

Ek (iX ; idS m ) ⊕ Z2

Ek (iY

Ek (iY

Ek (iY

Ek (iY

Ek (iY ; idS n ) ⊕ Z2

)

; idS n )

Furthermore, for m = 3, 7 and n = 1, we have

; idS n )

; idS n )

H.S. Shin et al. / Topology and its Applications 264 (2019) 382–393

Ek (iX ) ∼ = Ek (iY )

k ≥ m, 1

m>k≥1

1

Z2

391

Example 2. For an integer n > 1 and two commutative groups G and H, the Eilenberg-MacLane spaces K(G, n) and K(H, n − 1) are now considered. Then, [K(G, n) ∧ K(H, n − 1), K(G, n) × K(H, n − 1)] = 0 and [K(G, n), K(H, n − 1)] = 0. By Proposition 4.3, we have Ek (K(G, n)) ⊕ Ek (K(H, n − 1)) ∼ = Ek (K(G, n) × K(H, n − 1), K(H, n − 1)) for all k > 0. Proposition 5.1. Let (X, A) be a relative CW-complex and (f1 , f2 ) be a morphism from the inclusion i : A → X to itself. For given h1  f1 , there exists an extension h2 of h1 such that (h1 , h2 )  (f1 , f2 ). Proof. Let h1  f1 and Gt be a homotopy between h1 and f1 such that G0 = f1 and G1 = h1 . As (X, A) is a relative CW-complex, (X, A) has the homotopy extension property. Thus, there exists a homotopy Ht such that H0 |A = f1 and Ht |A = Gt . Let h2 = H1 . Then, h2 |A = h1 and h2 ◦ i = i ◦ h1 . Therefore, (h1 , h2 )  (f1 , f2 ). 2 Let now G1 and G2 be abelian groups, and let M1 = M (G1 , n1 ) and M2 = M (G2 , n2 ) be the corresponding Moore spaces. Let X = M1 ∨ M2 and ik : Mk → X be the inclusion map and pj : X → Mj be the projection for j, k = 1, 2. If f : X → X is a self-map, let fjk : Mk → Mj be defined by fjk = pj ◦ f ◦ ik for j, k = 1, 2. Example 3. Let M1 = M (Z2 , n + 1) and M2 = M (Z3 , n) be Moore spaces for n ≥ 5. Then M1 and M2 are n-connected and (n − 1)-connected spaces, respectively. Let X = M1 ∨ M2 and J = Mj for j = 1, 2. By Theorem 4.4, we have a split exact sequence 1

Ek (iJ ; idJ )

Ek (iJ )

Ek (J)

1

for k ≤ 2n. By [5, Theorem 4.4 and 4.9], it holds that Edim X (X) ∼ = Z2 and Edim X+1 (X) = 1, whereas by dim X dim X [2, Theorem 3.8], we have E (M1 ) ∼ (M2 ) = 1. Edim X (iJ ) is now determined. = Z2 and E Case 1. Let J = M1 . Then, Edim X (iJ ) ∼ = Edim X (iJ ; idJ ) ⊕ Z2 . Since Edim X (iJ ) is a subgroup of Edim X (X) and Edim X (X) ∼ = Z2 , we have Edim X (iJ ; idJ ) = 1 and Edim X (X) ∼ = Edim X (iJ ) ∼ = Z2 .

392

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Case 2. Let J = M2 . Then, Edim X (iJ ) = Edim X (ij ; idJ ). Edim X (ij ; idJ ) is now determined. For each [f ] ∈ Edim X (iJ ; idJ ), we have the commutative diagram i2

M2

M1 ∨ M 2

idM2

f

M2

i2

M1 ∨ M 2

As i2 ◦ idM2 = i2 = f ◦ i2 , pj ◦ f ◦ i2 =

 0 idM2

if j = 1, if j = 2.

By [2, Proposition 2.6], [5, Lemma 3.2 and 3.3], and Proposition 5.1, for each [f ] ∈ Edim X (iJ ; idJ ), [f ] can be identified with ([f11 ], [0], [0], [idM2 ]) for some [f11 ] ∈ Edim (M1 ). As Edim (M1 ) ∼ = Z2 , there exists a nong ] ∈ Edim (M1 ). Let [h] = ([¯ g ], [0], [0], [idM2 ]). As M1 is a retract of X, we have the following trivial element [¯ commutative diagram: 0

πk (M2 )

i2

πk (M2 )

πk (X, M2 )

g

idM2 

0

πk (X)

i2

πk (X)

0

g

πk (X, M2 )

0

for any k ≥ 0. ¯ ∨ idM ) = h ¯  ⊕ idM  and πk (X) ∼ Since h = (h = πk (M1 ) ⊕ πk (M2 ) for k ≤ 2n, it holds that idM2  2 2 and the first g are the identity homomorphisms on homotopy groups. Thus, the second g is the identity homomorphism. Hence, [h] ∈ Edim X (iJ ; idJ ). This implies that Edim X (iJ ; idJ ) ∼ = Z2 . Thus, Edim X (iJ ) = Edim X (X). Therefore, Edim X (iM1 ) ∼ = Edim X (iM2 ) ∼ = Z2 . However, Edim X (iM1 ; idM1 ) is not isomorphic to Edim X (iM2 ; idM2 ). References [1] M. Arkowitz, The Group of Self-Homotopy Equivalences a Survey, Lecture Notes in Math., vol. 1425, Springer, New York, 1990, pp. 170–203. [2] M. Arkowitz, K. Maruyama, Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topol. Appl. 87 (2) (1998) 133–154. [3] M. Arkowitz, O. Lupton, On Finiteness of Subgroups of Self-Homotopy Equivalences, Contemporary Mathematics, vol. 181, American Mathematical Society, 1995, pp. 1–25. [4] M. Arkowitz, H. Oshima, J. Strom, Homotopy classes of self-maps and induced homomorphisms of homotopy group, J. Math. Soc. Jpn. 58 (2) (2006) 401–418. [5] H. Choi, K. Lee, Certain self homotopy equivalences on wedge product on Moore spaces, Pac. J. Math. 272 (1) (2014) 35–57. [6] H. Choi, K. Lee, Certain numbers on the groups self-homotopy equivalences, Topol. Appl. 181 (2015) 104–111. [7] H. Choi, K. Lee, H. Oh, Self-homotopy equivalences related to cohomotopy groups, J. Korean Math. Soc. 54 (2017) 399–415.

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[8] H. Yu, W. Shen, The groups of self homotopy equivalences of topological pair, Acta Math. Chin. Ser. 42 (3) (2011) 521–528. [9] K. Lee, The groups of self pair homotopy equivalences, J. Korean Math. Soc. 43 (3) (2006) 491–506. [10] J. Rutter, Spaces of Homotopy Self-Equivalences - A Survey, Lecture Notes in Mathematics, vol. 1662, Springer-Verlag, Berlin, 1997, x+170 pp.