Self-homotopy equivalences of Moore spaces depending on cohomotopy groups

https://doi.org/10.4134/JKMS.j180680 pISSN: 0304-9914 / eISSN: 2234-3008

SELF-HOMOTOPY EQUIVALENCES OF MOORE SPACES DEPENDING ON COHOMOTOPY GROUPS

Ho Won Choi, Kee Young Lee, and Hyung Seok Oh

Abstract. Given a topological space X and a non-negative integer k, E

(X) is the set of all self-homotopy equivalences of X that do not change maps from X to an t-sphere S

homotopically by the composition for all t ≥ k. This set is a subgroup of the self-homotopy equivalence group E(X). We find certain homotopic tools for computations of E

(X). Using these results, we determine E

(M (G, n)) for k ≥ n, where M (G, n) is a Moore space type of (G, n) for a finitely generated abelian group G.

1. Introduction

For a topological space X, we denote E (X) as the set of all homotopy classes of self-homotopy equivalences of X. Then E (X) is a subset of [X, X] and has a group structure given by the composition of homotopy classes. The subset E(X) has been studied extensively by various authors, including Arkowitz [2], Arkowitz and Maruyama [3], Lee [5, 7], Rutter [9], Sawashita [10], and Sierad- ski [11]. Moreover several subgroups of E (X) have also been studied, notably the group E _] ^k (X), which consists of all elements of E (X) that induce the iden- tity homomorphism on homotopy groups π _t (X) for t = 0, 1, 2, . . . , k. In our previous work [4], the first and second authors used homotopy techniques to calculate these subgroups for the wedge products of Moore spaces.

In [6], we introduced E _k ^] (X), which consists of the elements of E (X) that induce the identity homomorphism on cohomotopy groups π ^t (X) for t ≥ k.

Equivalently, it can be defined as follows: For a non-negative integer i, consider the self-map f : X → X such that g ◦ f is homotopic to g for each g : X → S ^t

Received October 12, 2018; Accepted January 24, 2019.

2010 Mathematics Subject Classification. Primary 55P10, 55Q05, 55Q55.

Key words and phrases. self-homotopy equivalence, cohomotopy group, Moore space, co- Moore space.

The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of education(NRF- 2015R1C1A1A01055455).

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).

c

2019 Korean Mathematical Society

1371

and for each t ≥ k. The set of all homotopy classes of such self-maps of X is denoted by [X, X] ^] _k , that is,

[X, X] ^] _k = f ∈ [X, X] | g ◦ f ∼ g for each g : X → S ^t for all t ≥ k . Then

E _k ^] (X) = E (X) ∩ [X, X] ^] _k .

In [6], we proved that E _k ^] (X) is a subgroup of E (X) and, if X is a finite CW- complex, then E _k ^] (X) has a lower bound whereas E _] ^k (X) has an upper bound.

Moreover, we calculated E _k ^] (X) for special Moore space X = M (Z p , n) and co-Moore space X = C(Z p , n).

In this paper, we determine E _k ^] (M (G, n)) completely for k ≥ n, where M (G, n) is a Moore space type of (G, n) with G a finitely generated abelian group and n ≥ 3. To solve this problem, we first study a subset Z _k ^] (Y, Z) of [Y, Z] for spaces Y and Z. The subset Z _k ^] (Y, Z) is defined by the set of all h ∈ [Y, Z] whose induced homomorphism h ^]t : π ^t (Z) → π ^t (Y ) is the trivial homomorphism for t ≥ k. Furthermore, we investigate the properties of E _k ^] (X) for given wedge product space X to prove the following theorem in Section 4:

Theorem 4.3. Let M (G, n) be a Moore space type of (G, n) and G = F ⊕ T be a finitely generated abelian group G with free part F and torsion part T . Then E _k ^] (M (G, n)) is isomorphic to

E _k ^] (M (F, n)) ⊕ Z _k ^] (M (F, n), M (T, n)) ⊕ Z _k ^] (M (T, n), M (F, n)) ⊕ E _k ^] (M (T, n)).

From Theorem 4.3, the problem of computing E _k ^] (M (G, n)) reduces to that of computing the E _k ^] -groups and Z _k ^] -groups for Moore spaces for (possibly in- finite) cyclic groups. In Section 5, we compute explicitly Z _k ^] -groups for Moore spaces for (possibly infinite) cyclic groups and combine the relevant E _k ^] -groups computed in our previous paper and are recorded as Theorem 2.4 to obtain the following main result:

Corollary 5.13. Let G =

 _m L

i=1 Z

 L

s

L

j=1 Z ^q

!

. Then, we have

E _k ^] (M (G, n)) ∼ =



 

 

 

 



 

 

 

 



GL(m, Z)

m×t

L Z 2 s

L

j=1

 _m L Z q



L E(M (T, n)) if k ≥ n + 2,

GL(m, Z)

m×t+t+`

L Z 2 L

s

L

j=1

 _m L Z q

 !

if k = n + 1,

GL(m, Z)

t+`

L Z 2 L

s

L

j=1

 _m L Z q

 !

if k = n,

where GL(m, Z) is the general linear group of degree m, t is the number of even q j and ` is the number of pairs {i, j} ⊂ {1, . . . , s} such that both q i and q j are even and i 6= j.

Throughout this paper, all topological spaces are based and have the based homotopy type of a CW-complex, and all maps and homotopies preserve base points. For the spaces X and Y , we denote by [X, Y ] the set of all homotopy classes of maps from X to Y . No distinction is made between the notation of a map X → Y and that of its homotopy class in [X, Y ]. When a group G is generated by a set {a ₁ , . . . , a _n }, then we denote it by G{a ₁ , . . . , a _n }. More- over, when f : X → Y is a map, f ^]k : π ^k (Y ) → π ^k (X) denote the induced homomorphisms on the k-th cohomotopy group.

2. Preliminaries

In this section, we review some results provided in [3, 6], knowledge of which would be useful when reading this paper. First, we introduce the following proposition from [3] that is a basic concept in developing this paper.

Proposition 2.1 (Arkowitz and Maruyama [3]). If X is (k − 1)-connected, Y is (` − 1)-connected and, further, if k, ` ≥ 2 and dim P ≤ k + ` − 1, then the projections X ∨ Y → X and X ∨ Y → Y induce a bijection:

[P, X ∨ Y ] → [P, X] ⊕ [P, Y ].

Consider abelian groups G 1 and G 2 and Moore spaces M 1 = M (G 1 , n 1 ) and M 2 = M (G 2 , n 2 ). When X = M 1 ∨ M 2 , we denote by i j : M j → X the inclusion and by p j : X → M j the projection, where j = 1, 2. If f : X → X, then we define f jk = M k → M j by f jk = p j ◦ f ◦ i k for j, k = 1, 2. Then, by Proposition 2.1, we have

[X, X] ∼ = [M 1 , M 1 ] ⊕ [M 1 , M 2 ] ⊕ [M 2 , M 1 ] ⊕ [M 2 , M 2 ]

and, from [3, Proposition 2.6], there exists a bijective function θ that assigns to each f ∈ [X, X] a 2 × 2 matrix

θ(f ) = f 11 f 12

f 21 f 22

 ,

where f jk ∈ [M k , M j ]. In addition, we have the following:

(1) θ(f + g) = θ(f ) + θ(g), so θ is an isomorphism [X, X] → L

j,k=1,2

[M k , M j ];

(2) θ(f ◦g) = θ(f )θ(g), where f ◦g denotes composition in [X, X] and θ(f )θ(g) denotes matrix multiplication.

Moreover, for each f ∈ [X, X], the induced homomorphism f ^]k on the coho- motopy groups π ^k (X) is determined as in the following propositions:

Proposition 2.2 (Proposition 3.4 in [6]). For any f ∈ [M 1 ∨ M 2 , M 1 ∨ M 2 ], we have

f ^]k (γ 1 , γ 2 ) = (f ₁₁ ^]k (γ 1 ) + f ₂₁ ^]k (γ 2 ), f ₁₂ ^]k (γ 1 ) + f ₂₂ ^]k (γ 2 )),

where γ ₁ ∈ π ^k (M ₁ ) and γ ₂ ∈ π ^k (M ₂ ).

Proposition 2.3 (Proposition 3.5 in [6]). If f ∈ E _k ^] (M ₁ ∨ M 2 ), then f ^]k = 1 _π

(M

) 0

0 1 _π

(M

)

 .

In [6], we computed E _k ^] (M (Z ^q , n)) and obtained the following table:

Theorem 2.4 (Theorems 4.1, 4.2, 4.3, and 4.4 in [6]).

q : odd q ≡ 0 (mod 4) q ≡ 2 (mod 4)

E _n+1 ^] (M (Z q , n)) 1 Z 2 Z 2

E _n ^] (M (Z ^q , n)) 1 Z 2 Z 2

E _n−1 ^] (M (Z q , n)) 1 1 1

3. Maps inducing a trivial homomorphism on cohomotopy groups In [8], Maruyama studied the subset Z _] ^k (Y, Z) of [Y, Z]. Z _] ^k (Y, Z) is the sub- set of all homotopy classes from Y to Z that induce the trivial homomorphism π t (Y ) to π t (Z) for 0 ≤ t ≤ k. In this section, we introduce a subset Z _k ^] (Y, Z) of [Y, Z] that is a dual concept of Z _] ^k (Y, Z) and, in particular, investigate some properties of Z _k ^] (Y, Z) for wedge spaces Y and Z.

Definition 3.1. Let Y and Z be topological spaces. Then the subset Z _k ^] (Y, Z) is defined by the set of all h ∈ [Y, Z] whose induced homomorphism h ^]t : π ^t (Z) → π ^t (Y ) is the trivial homomorphism for t ≥ k. Equivalently,

Z _k ^] (Y, Z) = {f ∈ [Y, Z] | α ◦ f ' 0 for all α : Z → S ^t , t ≥ k}.

If Z = Y , then Z _k ^] (Y, Y ) is simply denoted by Z _k ^] (Y ).

It is well known that there is a bijective map τ : [Y ∨Z, W ] → [Y, W ]⊕[Z, W ] defined by τ (f ) = (f ◦ i _Y , f ◦ i _Z ), where i _I : I → Y ∨ W is an inclusion map for I = Y, W . The inverse of τ is defined by ρ : [Y, W ] ⊕ [Z, W ] → [Y ∨ Z, W ] defined by ρ(g, h) = ∇ ◦ (g ∨ h), where ∇ is the folding map.

Proposition 3.1. Let Y , Z, and W be CW-complexes. Then there is a bi- jective map τ : Z _k ^] (Y ∨ Z, W ) → Z _k ^] (Y, W ) ⊕ Z _k ^] (Z, W ) defined by τ (f ) = (f ◦ i _Y , f ◦ i _Z ).

Proof. It is sufficient to show that τ (Z _k ^] (Y ∨ W )) ⊂ Z _k ^] (Y, W ) ⊕ Z _k ^] (Z, W ) and ρ(Z _k ^] (Y, W ) ⊕ Z _k ^] (Z, W )) ⊂ Z _k ^] (Y ∨ Z, W ).

Let f ∈ Z _k ^] (Y ∨ Z, W ) and t ≥ k. Since f ^]t = 0, (f ◦ i I ) ^]t = i ^] _I ◦ f ^]t = 0 for I = Y, W . Hence, τ (f ) = (f ◦ i Y , f ◦ i Z ) ∈ Z _k ^] (Y, W ) ⊕ Z _k ^] (Z, W ).

Let (g, h) ∈ Z _k ^] (Y, W ) ⊕ Z _k ^] (Z, W ). Since g ^]t = 0 and h ^]t = 0 for t ≥ k, ρ(g, h) ^]t = (∇ ◦ (g ∨ h)) ^]t = (g ∨ h) ^]t ◦ ∇ ^]t = (g ^]t ∨ h ^]t ) ◦ ∇ ^]t = (0 ∨ 0) ◦ ∇ ^]t = 0.

Hence, ρ(g, h) = ∇ ◦ (g ∨ h) ∈ Z _k ^] (Y ∨ Z, W ). 

Define a map

Φ : [Y, W ∨ Z] → [Y, W ] ⊕ [Y, Z]

by Φ(f ) = (p W ◦ f, p Z ◦ f ), where p I = W ∨ Z → I is the projection map for I = W, Z. If Y is a suspension of a space, that is Y = ΣY ⁰ for some Y ⁰ , then there is a suspension co-multiplication C Y . Using C Y , we can obtain a map

Ψ : [Y, W ] ⊕ [Y, Z] → [Y, W ∨ Z]

given by Ψ(g, h) = (g ∨ h) ◦ C Y . Then Φ ◦ Ψ = id [Y,W ] ⊕ id [Y,Z] . Hence Ψ is an injection and Φ is a surjection.

Proposition 3.2. Let Y , Z, and W be CW-complexes. Suppose that Y = ΣY ⁰ with dim(Y ) ≤ k + ` − 1, Z is (k − 1)-connected, and W is (` − 1)- connected for k, ` ≥ 2. Then there is a bijection map from Z _k ^] (Y, Z ∨ W ) to Z _k ^] (Y, Z) ⊕ Z _k ^] (Y, W ).

Proof. By Proposition 2.1, Φ : [X, Y ∨ Z] → [X, Y ] ⊕ [X, Z] is bijective. Since Y = ΣY ⁰ , there is a suspension co-multiplication C _Y and the map Ψ : [X, Y ] ⊕ [X, Z] → [X, Y ∨ Z] is bijective. By a method similar to Proposition 3.1, we

can complete the proof. 

Let Y be a Moore space of type (G 1 ⊕ G 2 , n) for n ≥ 3 and let Y 1 and Y 2 be Moore spaces of type (G 1 , n) and (G 2 , n), respectively. Then we have Y ' Y 1 ∨ Y 2 .

Corollary 3.3. Let Y be a Moore space type of (G ₁ ⊕ G ₂ , n) for n ≥ 3. Then we have

Z _k ^] (Y ) ∼ = Z _k ^] (Y 1 ) ⊕ Z _k ^] (Y 1 , Y 2 ) ⊕ Z _k ^] (Y 2 , Y 1 ) ⊕ Z _k ^] (Y 2 ), where Y 1 = M (G 1 , n) and Y 2 = M (G 2 , n).

4. Properties of E _k ^] (M (G, n))

In [6], we investigated some properties and determined E _k ^] (M (Z ^p , n)). In this section, we extend and apply these results to a Moore space M (G, n), where G is a finitely generated abelian group and n ≥ 3. Since G is a finitely generated abelian group, G = L s

i=1 G _i , where G _i is a cyclic group. Then, we have M (G, n) ' W s

i=1 M (G _i , n). Therefore, we have [M (G, n), M (G, n)] ∼ =

s

M

i,j=1

[M (G i , n), M (G j , n)]

and

π ^k (M (G, n)) ∼ =

s

M

i=1

π ^k (M (G i , n)) by Proposition 2.1.

Let i i = M (G i , n) → M (G, n) be the inclusion and let p j : M (G, n) →

M (G j , n) be the projection. For a self-map f : M (G, n) → M (G, n), we define

f _ji = M _i → M _j by f _ji = p _j ◦ f ◦ i _i . Then there is a bijection θ that assigns each f ∈ [M (G, n), M (G, n)] the s × s matrix

θ(f ) =







f 11 · · · f 1s

.. . . . . .. . f _s1 · · · f _ss





 . Thus, there is a bijection θ ^] that assigns each

f ^]k ∈ Hom(π ^k (M (G, n)), π ^k (M (G, n))) the s × s matrix

θ ^] (f ^]k ) =







f ₁₁ ^]k · · · f _1s ^]k .. . . . . .. . f _s1 ^]k · · · f _ss ^]k





 .

Throughout this paper, θ(f ) and θ ^] (f ^]k ) are identified with f and f ^]k and are called the matrix representations of f and f ^]k , respectively. Furthermore, each γ ∈ π ^k (M (G, n)) can be represented as γ = (γ ₁ , γ ₂ , . . . , γ _s ), where γ _i = γ ◦ i _i ∈ π ^k (M (G _i , n)).

Proposition 4.1. For each f ∈ [M (G, n), M (G, n)], we have f ^]k (γ) = (

s

X

i=1

f _i1 ^]k (γ i ), . . . ,

s

X

i=1

f _is ^]k (γ i )) for γ ∈ π ^k (M (G, n)) and γ i = γ ◦ i i .

Proof. This can be proved in a manner similar to Proposition 3.4 in [6].  The matrix representations of the identity map 1 M (G,n) and the induced map 1 ^]t _{M (G,n)} are

θ(1 _{M (G,n)} ) =













1 _M

0 · · · 0 0 1 _M

.. . .. . . . . 0 0 · · · 0 1 M











 and

θ ^] (1 ^]t _{M (G,n)} ) =











 1 ^]t _M

1

0 · · · 0 0 1 ^]t _M

2

.. . .. . . . . 0 0 · · · 0 1 ^]t _M

s











 ,

respectively, where 1 _M

is the identity map in [M (G _i , n), M (G _i , n)] and 1 ^]t _M

i

is the induced map of 1 M

on the t-th cohomotopy group. θ(1 _{M (G,n)} ) and

θ ^] (1 ^]t _{M (G,n)} ) are simply denoted by I s and I _s ^]t , respectively.

Proposition 4.2. Let G = L s

j=1 G _j be a finitely generated abelian group. If f ∈ E _k ^] (M (G, n)), then

f ^]t = I _s ^]t for t ≥ k.

Proof. For any f ∈ E _k ^] (M (G, n)), f induces the identity homomorphism on π ^t (M (G, n)) for t ≥ k. Thus f ^]t = id _π

(M (G,n)) = 1 ^]t _{M (G,n)} . Hence, θ ^] (f ^]t ) = θ ^] (1 _{M (G,n)} ) ^]t ) = I _s ^]t . Therefore,

f ^]t = I _s ^]t

for t ≥ k. 

We determine the set of self-homotopy equivalences that induce the identity map on cohomotopy groups for Moore space of type (G, n), where G is a finitely generated abelian group and n is a positive integer. G can be represented as follows:

G ∼ =

m

M

i=1

Z

!

⊕





s

M

j=1

Z q



 ,

where Z ^q

is a primary cyclic group and q j represents powers of prime numbers.

Let G 1 = ( L m i=1 Z)⊕

 L s j=1 Z q



and G 2 = ( L r i=1 Z)⊕

 L t j=1 Z q

 . Then,

(4.1) G ₁ ⊕ G 2 ∼ =

m+r

M

i=1

Z

! M





s+t

M

j=1

Z q



 ,

where q j+s = q _j ⁰ for 1 ≤ j ≤ t. Let X = M (G 1 ⊕ G 2 , n) be a Moore space.

Since

M (G 1 ⊕ G 2 , n) ' M (G 1 , n) ∨ M (G 2 , n), we have

(4.2) X ' (∨ ^m _i=1 M (Z, n)) ∨ ∨ ^s j=1 M (Z ^q

, n)
. From (4.2), we have

[X, X]

∼ = [∨ ^m _i=1 M (Z, n), ∨ ^m i=1 M (Z, n)] M ∨ ^s _j=1 M (Z ^q

, n), ∨ ^m _i=1 M (Z, n) 

M ∨ ^m _i=1 M (Z, n), ∨ ^s j=1 M (Z ^q

, n) M ∨ ^s _j=1 M (Z ^q

, n), ∨ ^s _j=1 M (Z ^q

, n) . For each f ∈ [X, X], the matrix representation of f is

θ(f ) =







p 1 ◦ f ◦ i 1 · · · p m+s ◦ f ◦ i 1

.. . . . . .. . p ₁ ◦ f ◦ i m+s · · · p _m+s ◦ f ◦ i m+s







and we have f ^]k (γ) =

m+s

X

i=1

(p i ◦ f ◦ i 1 ) ^]k (γ i ), . . . ,

m+s

X

i=1

(p i ◦ f ◦ i m+s ) ^]k (γ i )

! , where γ i = γ ◦ i i .

Now, we divide the matrix representation of f into θ(f ) =

 M ₁ (f ) M ₂ (f ) M 3 (f ) M 4 (f )

 ,

where M 1 (f ) is the square matrix of degree n whose components are the maps in [M (Z, n), M (Z, n)], M ² (f ) is the m × s matrix whose components are the maps in [M (Z, n), M (Z ^q

, n)], M 3 (f ) is the s × m matrix whose components are the maps in [M (Z q

, n), M (Z, n)], and M 4 (f ) is the square matrix of degree s matrix whose components are the maps in [M (Z q

, n), M (Z q

, n)].

From the above matrix, we have the following induced matrix:

θ ^] (f ^]t ) =

 M 1 (f ) ^]t M 2 (f ) ^]t M ₃ (f ) ^]t M ₄ (f ) ^]t

 . In [3], for given X = M (F ⊕ T, n),

E(X) ∼ = E (M (F, n)) ⊕ [M (F, n), M (T, n)] ⊕ [M (T, n), M (F, n)]

(4.3)

⊕ E(M (T, n)),

where F and T are finitely generated abelian groups.

Theorem 4.3. Let X = M (F ⊕ T, n) be a Moore space with finitely generated abelian groups F and T and positive integer n. Then

E _k ^] (X) ∼ = E _k ^] (M (F, n)) ⊕ Z _k ^] (M (F, n), M (T, n))

⊕ Z _k ^] (M (T, n), M (F, n)) ⊕ E _k ^] (M (T, n)).

Proof. From (4.3), for f ∈ E (X), f can be represented by f = (f ₁ , f ₂ , f ₃ , f ₄ ) for some f 1 ∈ E(M (F, n)), f 2 ∈ [M (F, n), M (T, n)], f 3 ∈ [M (T, n), M (F, n)], and f 4 ∈ E(M (T, n)). Then,

M 1 (f ) = θ(f 1 ), M 2 (f ) = θ(f 2 ), M 3 (f ) = θ(f 3 ), and M 4 (f ) = θ(f 4 ).

Since, for t ≥ k,

θ ^] (f ^]t ) =

 M ₁ (f ) ^]t M ₂ (f ) ^]t M 3 (f ) ^]t M 4 (f ) ^]t



= I _m×s ^]t by Proposition 4.2,

M ₁ (f ) ^]t = I _m ^]t , M ₂ (f ) ^]t = O ^]t _m×s , M ₃ (f ) ^]t = O _s×m ^]t , M ₄ (f ) ^]t = I _s ^]t

for t ≥ k, where O ^]t _m×s and O ^]t _s×m are an m × s zero matrix and an s × m

zero matrix, respectively. Thus, if f = (f 1 , f 2 , f 3 , f 4 ) ∈ E _k ^] (X), then f 1 ∈

E _k ^] (M (F, n)), f ₂ ∈ Z _k ^] (M (F, n)), f ₃ ∈ Z _k ^] (M (F, n)), and f ₄ ∈ E _k ^] (M (T, n)).

Therefore, E _k ^] (X) is contained in

E _k ^] (M (F, n)) ⊕ Z _k ^] (M (F, n), M (T, n)) ⊕ Z _k ^] (M (T, n), M (F, n)) ⊕ E _k ^] (M (T, n)).

Conversely, let f = (f 1 , f 2 , f 3 , f 4 ) belong to

E _k ^] (M (F, n)) ⊕ Z _k ^] (M (F, n), M (T, n)) ⊕ Z _k ^] (M (T, n), M (F, n)) ⊕ E _k ^] (M (T, n)).

Then f ∈ E (X) from (4.3). Since θ ^] (f ₁ ^]t ) = I _m ^]t , θ ^] (f ₂ ^]t ) = O _s×m ^]t , θ ^] (f ₃ ^]t ) = O ^]t _m×s , and θ ^] (f ₄ ^]t ) = I _s ^]t , θ ^] (f ^]t ) = I _s×m ^]t for each t ≥ k. Therefore, f ∈

E _k ^] (X). 

5. Computations of E _k ^] (M (G, n)) Let X 1 = M ( L m

i=1 Z, n) and X ² = M ( L s

j=1 Z ^q

, n). For the wedge product space X = X 1 ∨ X 2 , to determine E _k ^] (X) by Theorem 4.3, we need to calculate each E _k ^] (X 1 ), Z _k ^] (X 2 , X 1 ), Z _k ^] (X 1 , X 2 ), and E _k ^] (X 2 ).

We first compute E _k ^] (X ₁ ). Since X ₁ ' ∨ ^m _i=1 M (Z, n) ' ∨ ^m i=1 S ⁿ _i , where S _i ⁿ is a copy of S ⁿ , we have

[X 1 , X 1 ] ∼ =

m

M

i,j=1

[S ⁿ _i , S _j ⁿ ] ∼ =

m×m

M

i=1

Z

by Proposition 2.1 in [9]. Thus, E (X 1 ) = GL(m, Z), where GL(m, Z) is the general linear group of degree m.

Let i i : S _i ⁿ → X 1 be the inclusion and p j : X 1 → S _j ⁿ be the projection. For a self-map f : X 1 → X 1 , we define f ji : S _i ⁿ → S _j ⁿ by f ji = p j ◦f ◦i i . Since S _i ⁿ and S ⁿ _j are the copies of the n-dimensional sphere, we see that [S _i ⁿ , S _j ⁿ ] = [S ⁿ , S ⁿ ].

Let ι n be the identity map on [S ⁿ , S ⁿ ]; in particular, let (ι n ) ji be the identity map on [S _i ⁿ , S ⁿ _j ]. Then the matrix representation of f is given by

θ(f ) =







t 11 (ι n ) 11 · · · t 1m (ι n ) 1m

.. . . . . .. . t m1 (ι n ) m1 · · · t mm (ι n ) mm





 , where t _ji is the degree of f _ji for j, i = 1, 2, . . . , m.

Lemma 5.1. For n ≥ 1,

Z _k ^] (S ⁿ ) ∼ = (

Z if k > n, 0 if k ≤ n.

Proof. If k > n, then [S ⁿ , S ^k ] = 0. Thus, each f ∈ [S ⁿ , S ⁿ ] induces the trivial homomorphism on π ^t (S ⁿ ) for t ≥ k. Hence, Z _k ^] (S ⁿ ) ∼ = Z. If k = n, then [S ⁿ , S ⁿ ] ∼ = Z{ι n }. Since f = (deg f )ι n for each f ∈ [S ⁿ , S ⁿ ], f ^]n (ι n ) = ((deg f )ι _n ) ^]n (ι _n ) = (deg f )ι _n . Thus, if f ∈ Z _k ^] (S ⁿ ), then deg f must be 0.

Hence, Z _n ^] (S ⁿ ) = 0. From the definition, we have Z _k ^] (S ⁿ ) = 0 for k < n. 

Theorem 5.2. For n ≥ 2, E _k ^] (X 1 ) ∼ =

( GL(m, Z) k > n,

1 k ≤ n.

Proof. Since π ^k (X 1 ) = 0 for k > n, E _k ^] (X 1 ) = E (X 1 ) for k > n. Suppose that k ≤ n. Then, for each f ∈ E _k ^] (X ₁ ), θ ^] (f ^]t ) = I _s ^]t for t ≥ k. Thus

f _ji ^]t =

( (ι _n ) ^]t _ji if i = j, 0 if i 6= j.

Hence, if i = j, then f _ji ∈ E _k ^] (S ⁿ ) and if i 6= j, then f _ji ∈ Z _k ^] (S ⁿ ). By Lemma 5.1, θ ^] (f ^]t ) = id ^]t _X

1

for t ≥ k. 

Now, we investigate Z _k ^] (X ₂ , X ₁ ) and Z _k ^] (X ₁ , X ₂ ). We review briefly the following lemmas in [1] and [4].

Lemma 5.3. Let M (Z ^q , n) be a Moore space type of (Z ^q , n). Then the k-th cohomotopy groups π ^k (M (Z q , n)) are isomorphic to

k ≥ n + 2 k = n + 1 k = n

q ≡ 1 (mod 2) 0 Z q 0

q ≡ 0 (mod 2) 0 Z q Z 2

Generator - ι _n+1 ◦ π _q η _n ◦ π _q

Lemma 5.4. Let M (Z ^q , n) be a Moore space type of (Z ^q , n). Then the k-th homotopy groups π _k (M (Z q , n)) are isomorphic to

k = n + 2 k = n + 1 k = n k ≤ n − 1

q ≡ 1 (mod 2) 0 0 Z q 0

q ≡ 0 (mod 4) Z 2 ⊕ Z 2 Z 2 Z q 0

q ≡ 2 (mod 4) Z 4 Z 2 Z q 0

Generator - i _q ◦ η n i _q ◦ ι n - Proposition 5.5. For k > 0,

Z _k ^] (M (Z q , n), M (Z, n)) ∼ =



 

 

0 if q ≡ 1 (mod 2),

Z 2 {η n ◦ π q } if k ≥ n + 1 and q ≡ 0 (mod 2), 0 if k ≤ n and q ≡ 0 (mod 2).

Proof. By Lemma 5.3,

[M (Z ^q , n), M (Z, n)] = π ⁿ (M (Z ^q , n)) ∼ =

( 0 if q ≡ 1 (mod 2), Z ² {η n ◦ π q } if q ≡ 0 (mod 2).

If q is odd, then Z _k ^] (M (Z q , n), M (Z, n)) = 0. Let q be even. If k ≥ n + 1,

then π ^k (M (Z, n)) = 0. Thus Z k ^] (M (Z ^q , n), M (Z, n)) = [M (Z ^q , n), M (Z, n)] =

π ⁿ (M (Z q , n)) ∼ = Z 2 . If k = n, then π ⁿ (M (Z, n)) ∼ = Z{ι n } and π ⁿ (M (Z q , n)) ∼ = Z ² {η n ◦ π q } and so

(η n ◦ π q ) ^]n (ι n ) = ι n ◦ η n ◦ π q = η n ◦ π q 6= 0.

Therefore, Z _n ^] (M (Z ^q , n), M (Z, n)) = 0. 

Proposition 5.6. For k ≥ n,

Z _k ^] (M (Z, n), M (Z ^q , n)) ∼ = Z q {i q ◦ ι n }.

Proof. By Lemma 5.4, [M (Z, n), M (Z q , n)] = π _n (M (Z q , n)) ∼ = Z q {i _q ◦ ι _n }.

Since π ^k (S ⁿ ) = 0 for k ≥ n + 1, Z _k ^] (M (Z, n), M (Z q , n)) ∼ = Z q . If k = n, then π ⁿ (S ⁿ ) ∼ = Z{ι n } and

π ⁿ (M (Z ^q , n)) ∼ =

( 0 if q ≡ 1 (mod 2), Z 2 {η n ◦ π q } if q ≡ 0 (mod 2).

If q ≡ 1 (mod 2), then Z _n ^] (M (Z, n), M (Z q , n)) ∼ = Z q . If q ≡ 0 (mod 2), then (i q ◦ ι n ) ^]n (η n ◦ π q ) = η n ◦ π q ◦ i q ◦ ι n = 0 because π q ◦ i q ' 0. Thus,

Z _n ^] (M (Z, n), M (Z q , n)) ∼ = Z q . 

Theorem 5.7. For k > 0, Z _k ^] (X ₂ , X ₁ ) ∼ =







m×t

L Z 2 if k ≥ n + 1,

0 if k ≤ n,

where t is the number of even q j . Proof. By Corollary 3.3,

Z _k ^] (X 2 , X 1 ) ∼ =

m

M

s

M

i=1

Z _k ^] (M (Z ^q

, n), M (Z, n))

! .

By Proposition 5.5, we have Z _k ^] (X 2 , X 1 ) ∼ =

m×t

L Z 2 for k ≥ n + 1, where t is the number of even q _j . If k ≤ n, then Z _k ^] (M (Z q

, n), M (Z, n)) = 0. Therefore,

Z _k ^] (X ₂ , X ₁ ) = 0. 

Theorem 5.8. For k ≥ n,

Z _k ^] (X 1 , X 2 ) ∼ =

s

M

j=1 m

M Z q

! .

Proof. This follows immediately from Corollary 3.3 and Proposition 5.6.  Finally, we determine E _k ^] (X 2 ). Since X 2 ∼ = ∨ ^s _i=1 M (Z ^q

, n), we have

[X 2 , X 2 ] ∼ =

s

M

j=1 s

M

i=1

[M (Z ^q

, n), M (Z ^q

, n)]

!

by Proposition 2.1. In [3], it was shown that E(X 2 ) ∼ =

s

M

i=1

E(M (Z q

, n))

! M



 M

i6=j

[M (Z q

, n), M (Z q

, n)]



 . Theorem 5.9. For k > 0,

E _k ^] (X 2 ) ∼ =

s

M

i=1

E _k ^] (M (Z ^q

, n))

! M



 M

i6=j

Z _k ^] (M (Z ^q

, n), M (Z ^q

, n))



 . Proof. Let f ∈ E _k ^] (X 2 ). Then θ ^] (f ^]t ) = I _s ^]t for t ≥ k. This means that f _ii ^]t = 1 _π

(M (Z

,n)) for all 1 ≤ i ≤ s and θ ^] (f _ji ^]t ) = 0 for i 6= j. Thus, f _ii ∈ E _k ^] (M (Z ^q

, n)) and f ji ∈ Z _k ^] (M (Z ^q

, n), M (Z ^q

, n)). Therefore,

f ∈ 

E _k ^] (M (Z q

, n))  M



 M

i6=j

Z _k ^] (M (Z q

, n), M (Z q

, n))



 .

Conversely, let f ∈ 

E _k ^] (M (Z q

, n)) 

L L

i6=j

Z _k ^] (M (Z q

, n), M (Z q

, n))

! . Then,

f _ji ^]t =

( 1 _π

(M (Z

,n)) if i = j,

0 if i 6= j,

for all t ≥ k. By Proposition 4.2 and the definition of E _k ^] (M (Z ^q

, n)), the matrix representation of f ^]t is equal to I _s ^]t for all t ≥ k. Hence, f ∈ E _k ^] (X ₂ ). 

From [3], we have the following lemma:

Lemma 5.10. Let M _j = M (Z q

, n) and M _i = M (Z q

, n). Then we have either q j or q i : odd q j ≡ q _i ≡ 2 (mod 4) q j ≡ q _i ≡ 0 (mod 4)

[M j , M i ] Z d Z 2d Z d ⊕ Z 2

Generator α j α j α j , i q ◦ η _n ◦ π _q

where π _q

◦ α j = ¯ jι _n+1 ◦ π q

, ¯ j is an integer such that q _j = ¯ jd, and d = (q _i , q _j ) is the greatest common divisor.

Proposition 5.11. Let M _i = M (Z q

, n) and M _j = M (Z q

, n). Then we have

Z _k (M _j , M _i ) ∼ =



 

 

[M _j , M _i ] if k ≥ n + 2,

0 if q j or q i : odd and k = n or n + 1, Z 2 if q _j ≡ q i ≡ 0 (mod 2) and k = n or n + 1.

Proof. If k ≥ n + 2, then π ^k (M j ) = π ^k (M i ) = 0. Thus, Z _k ^] (M j , M i ) = [M j , M i ] for k ≥ n + 2. By Lemma 5.3, we have π ⁿ⁺¹ (M ` ) ∼ = Z ^q

{ι n+1 ◦ π q

} and

π ⁿ (M _` ) ∼ =

( 0 if q ` ≡ 1 (mod 2),

Z 2 {η n ◦ π q

} if q _` ≡ 0 (mod 2),

where ` = i, j. By Lemma 5.10, we have

[M j , M i ] ∼ =



 

 

Z d {α j } if q j or q i : odd, Z 2d {α j } if q j ≡ q i ≡ 2 (mod 4), Z ^d ⊕ Z 2 {α j , i q

◦ η n ◦ π q

} if q j ≡ q i ≡ 0 (mod 4), where d = (q j , q i ).

Case 1. Let q j or q i be odd.

Let g ∈ [M _j , M _i ]. Then, g = sα _j for some 0 ≤ s ≤ d. Thus, we have g ^]n+1 (ι n+1 ◦ π q

) = ι n+1 ◦ π q

◦ sα j

= sι _n+1 ◦ π q

◦ α j

= sι _n+1 ◦ ¯ jι _n ◦ π q

= s¯ jι n ◦ π q

.

Since 0 ≤ s¯ j ≤ q j , g ^]n+1 is trivial if and only if s = 0. Hence, Z _n+1 ^] (M j , M i ) = 0. Moreover, Z _n ^] (M _j , M _i ) = 0 by the definition.

Case 2. Let q _j ≡ q _i ≡ 2 (mod 4).

Let g ∈ [M _j , M _i ]. Then g = sα _j for some 0 ≤ s ≤ 2d. Thus, if s = d, then g ^]n+1 (ι n+1 ◦ π q

) = ι n+1 ◦ π q

◦ i q

◦ η n ◦ π q

= 0

because π q

◦ i q

' 0. If s 6= d, then

g ^]n+1 (ι n+1 π q

) = sι n+1 π q

◦ α j = s¯ jι n+1 ◦ π q

.

Hence, s = 0 or d. Therefore Z _n+1 ^] (M _j , M _i ) ∼ = Z 2 {dα _j }. Then, for g ∈ Z _n+1 ^] (M _j , M _i ),

g ^]n+1 (η n ◦ π q

) = dη n ◦ π q

◦ α j = d¯ jη n+1 ◦ π q

= q j η n+1 ◦ π q

= 0 because q j is even. Therefore, Z _n ^] (M j , M i ) ∼ = Z 2 {dα j }.

Case 3. Let q j ≡ q i ≡ 0 (mod 4).

Let g ∈ [M j , M i ]. Then, g = sα j ⊕ ti q

◦ η n ◦ π q

for some 0 ≤ s < d and t = 0, 1. Then,

g ^]n+1 (ι n+1 ◦ π q

) = s¯ jι n+1 ◦ π q

⊕ tι n+1 ◦ π q

◦ i q

◦ η n ◦ π q

= s¯ jι n+1 ◦ π q

⊕ 0 because π _q

◦ i _q

' 0. Thus, s = 0 and t = 0, 1. Hence, Z _n+1 ^] (M _j , M _i ) ∼ = Z 2 {0 ⊕ i q

◦ η n ◦ π q

}. Then, for g ∈ Z _n+1 ^] (M _j , M _i ),

g ^]n (η n ◦ π q

) = 0 ⊕ η n ◦ π q

◦ i q

◦ η n ◦ π q

= 0.

Hence, Z _n ^] (M j , M i ) ∼ = Z 2 {0 ⊕ i q

◦ η n ◦ π q

}.  Theorem 5.12. For n ≥ 3,

E _k ^] (X ₂ ) ∼ =



 



 



 _s L

i=1 E(M (Z q

, n))



L L

i6=j

[M (Z q

, n), M (Z q

, n)]

!

if k ≥ n + 2,

t+` L

Z 2 if k = n or n + 1,

where t is the number of even q _j and ` is the number of pairs {i, j} ⊂ {1, . . . , s}

such that both q i and q j are even and i 6= j.

Proof. If k ≥ n + 2, then π ^k (X 2 ) = 0. Thus, E _k ^] (X 2 ) = E (X 2 ). If k = n or n + 1, then by [6, Theorems 4.1 and 4.2], E _k ^] (M (Z ^q

, n)) ∼ =

t

L Z 2 , where t is the number of even q i . By Proposition 5.11, L

i6=j

Z _k ^] (M (Z ^q

, n), M (Z ^q

, n)) ∼ =

`

L Z 2 , where ` is the number of pairs {i, j} ⊂ {1, . . . , s} such that both q i and q j are even and i 6= j. Therefore,

E _k ^] (X 2 ) ∼ =

t+`

M

Z 2 . 

If we combine Theorems 5.2, 5.7, 5.8, and 5.12, we obtain the following main result:

Corollary 5.13. Let G =

 _m L

i=1

Z

 L

s

L

j=1

Z q

!

. Then, we have

E _k ^] (M (G, n)) ∼ =



 

 

 

 



 

 

 

 



GL(m, Z)

m×t

L Z 2 s

L

j=1

 _m L Z q



L E(M (T, n)) if k ≥ n + 2,

GL(m, Z)

m×t+t+`

L Z 2 L L ^s

j=1

 _m L Z q

 !

if k = n + 1,

GL(m, Z)

t+`

L Z 2 L

s

L

j=1

 _m L Z q

 !

if k = n, where GL(m, Z) is the general linear group of degree m, t is the number of even q j and ` is the number of pairs {i, j} ⊂ {1, . . . , s} such that both q i and q j are even and i 6= j.

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equivalences and related topics (Montreal, PQ, 1988), 170–203, Lecture Notes in Math., 1425, Springer, Berlin, 1990. https://doi.org/10.1007/BFb0083840

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http://projecteuclid.org/euclid.pjm/1102971956

Ho Won Choi

Institute of Natural Science Korea University

Sejong 339-700, Korea

Email address: [email protected]

Kee Young Lee

Division of Applied Mathematical Sciences Korea University

Sejong 339-700, Korea

Email address: [email protected]

Hyung Seok Oh

Department of Mathematics Korea University

Seoul 702-701, Korea

Email address: [email protected]