Path-components of the Mapping Space-I

Path-components of the Mapping Space-I

Marek GOLASI ´NSKI^* and Akira KONO^**

(Received October 16, 2016)

We make use of the evaluation fibration to describe homopoty groups of path-components of the mapping space Map(S^m, M(A, n)) from them-sphereS^mto the Moore spaceM(A, n) of type (A, n).

Homotopy types of path-components of the spaces Map(M(Zk, m), M(Zk, n)) and Map(M(Zk, m),Sⁿ) are studied as well.

Key words_：compact-open topology, mapping space, Moore space, path-component.

1. Introduction

Let X and Y be connected spaces and let Map(X, Y) denote the space of all continuous (not necessarily based) maps between X and Y with the compact-open topology. The space Map(X, Y) is gen- erally disconnected with path-components in one-to- one correspondence with the set

X, Y

of (free) ho- motopy classes of maps. Furthermore, different com- ponents may—and frequently do—have distinct ho- motopy types. A basic problem in homotopy the- ory is to determine whether two components are ho- motopy equivalent or, more generally, to classify the path-components of Map(X, Y) up to homotopy type.

For a basepoint x0∈X, we have the evaluation map ω: Map(X, Y)→Y, defined byω(g) =g(x0), forg∈ Map(X, Y) which is a fibration. Let Map_f(X, Y) de- note the path-component of Map(X, Y) that contains a given mapf:X →Y and writeω_f: Map_f(X, Y)→Y for the restriction of the mapω to Map_f(X, Y).

Works on these classification problems date back to the 1940’s. Whitehead²¹⁾ Theorem 2.8 consid-

* Faculty of Mathematics and Computer Science, University of Warmia and Mazury, S loneczna 54 Street, 10-710 Olsztyn, Poland

E-mail : [email protected]

** Faculty of Science and Engineering, Doshisha University, 1-3 Tatara, Miyakodani, Kyotanabe-shi, Kyoto, 610-0394 Japan

E-mail : [email protected]

ered the case X = S^m and Y = Sⁿ, in which a path-component corresponds to f ∈ πm(Sⁿ), and proved that Map_f(S^m,Sⁿ) is homotopy equivalent to M0(S^m,Sⁿ) if and only if the evaluation fibrationωf : Map_f(S^m,Sⁿ)→ Sⁿ admits a section. Next, Hansen

9, 10), Koh¹¹⁾and later McClendon¹⁵⁾, extended this approach. In Golasi´nski⁵⁾the authors make use of Got- tlieb groups of spheres to deal with path-components of the spaces Map(S^n+k,Sⁿ) for 8≤k≤13.

Let G be a Lie group and P → X a princi- pal G-bundle over a space X with gauge group G. Then by Atiyah and Bott ²⁾ we have: Let BG be the classifying space for G. Then in homotopy theory BG =M_P(X, BG). Here the subscript P denotes the path-component of a map ofX intoBGwhich induces P.

The case in whichX is a manifold andY =BG has been the subject of extensive recent research by Crabb, Kono, Sutherland, Tsukuda and others. (See e.g., Crabb and Sutherland⁴⁾, Kono and Tsukuda¹²⁾, Sutherland¹⁸⁾.)

Then, Lupton and Smith¹⁴⁾give a general

method that may be effectively applied to the ques- tion of whether two components of a function space Map(X, Y) have the same homotopy type providedX is a co-H-space.

The main advantage that we take in this paper is to classify homotopy types of path- components of the mapping spaces Map(S^m, M(B, n)), Map(M(A, m), M(B, n)) and Map(M(A, m), Sⁿ), whereS^kis thek-sphere andM(A, l) the Moore space of type (A, l) for an abelian groupAandl≥2.

The aim of Section 2. is to fix some notations, recall definitions and necessary results. In particular, a very important result onωf : Map_f(S^m, X)→Xfor our further investigations which was obtained by G.W.

Whitehead²¹⁾ with a correction by J.H.C. Whitehead

22) and then generalized by Lang¹³⁾ Theorem 2.6 for ωf : Map_f(EA, X)→ X, whereEA denotes the sus- pension of the based spaceA.

Next, Section 3. makes use of some results from Golasi´nski and Mukai⁶⁾Chapter 1 and follows Koh¹¹⁾ to expounds path-components of Map(S^m, M(A, n)) for particularm, nand describe their homotopy groups in Proposition 3.1.

Finally, Section 4., based mainly on Araki and Toda¹⁾, is devoted to path-components of the spaces Map(M(Z^k, m),Sⁿ) and Map(M(Z^k, m), M(Z^k, n)) for the cyclic groupZk of orderkandm=n−1, n.

2. Prerequisites

For topological spacesX and Y, let Map(X, Y) be the space of all continuous maps equipped with the compact-open topology. In the pointed case for this space we write Map_∗(X, Y). Let Map_f(X, Y) (resp.

M_∗f(X, Y)) be the path-component of Map(X, Y) (resp. Map_∗(X, Y)) containing a (resp. pointed) map f :X→Y and denote byM0(X, Y) (resp.M_∗0(X, Y)) the one containing the constant map.

Given pointed spaces X, Y write [X, Y] and X, Yfor the sets of homotopy classes of pointed and free maps, respectively. It is well known that that there is an action of the fundamental groupπ₁(Y) on [X, Y]

and there is a bijectionX, Y ≈[X, Y]/π1(Y).

Let [f] denote the free homotopy class of a map f : X → Y. If X is a Hausdorff space then for any homotopyH :I×X →Y there is an associated con- tinuous map H : I → Map(X, Y) and consequently [f]⊆Map_f(X, Y).

If the evaluation map ev : Map(X, Y)×X →Y is continuous and σ : I → Map(X, Y) is a path then its adjoint ˆσ:I×X −−−−→^σ×idX Map(X, Y)×X−→^ev Y is also continuous and so [f]⊇Map_f(X, Y).

In particular, ifXis a compactly generated space then for any pathσ:I→Map(X, Y) its adjoint ˆσ:I× X→Y is continuous and [f]⊇Map_f(X, Y). Because X is also Hausdorff we derive that [f] = Map_f(X, Y) and the set X, Yof homotopy classes of free maps X→Y coincides with the set of all path-components of the space Map(X, Y).

As pointed out by Whitehead ²¹⁾ all path- components of Map_∗(Sⁿ, X) for the n-sphere Sⁿ have the same homotopy type. Moreover, Lang ¹³⁾ Lemma 2.1 has generalized this result for the space Map_∗(EA, X), whereEAis the reduced suspension of the pointed space A. But, in general, distinct path- components of the space Map_∗(X, Y) need not be ho- motopy equivalent.

Therefore the following problem naturally arises.

Problem 2.1 Given spaces X and Y, classify all path-components of the space Map(X, Y)up to homo- topy type.

Throughout the rest of this paper, all spaces are assumed to be pointed compactly generated, all maps are pointed maps and homotopies are base-point pre- serving. Further, we do not distinguish between a map and its homotopy class and we use freely notations from the Toda’s book¹⁹⁾.

Given a pointed space X, the suspension EA of a pointed space A and f ∈ [EA, X] there is the evaluation fibration (at the base point of EA) Map_∗f(EA, X) → Map_f(EA, X) −−→^ωf X. A very im- portant result for our further investigations was ob- tained by G.W. Whitehead²¹⁾ with a correction by

J.H.C. Whitehead²²⁾and then generalized by Lang¹³⁾ Theorem 2.6. It describes the boundary operator in the homotopy sequence for the evaluation fibration by the generalized Whitehead product.

Theorem 2.2 (G. W. Whitehead) If f ∈[EA, X]

then for any spaceB andi≥1there is a commutative diagram

[E(A∧Eⁱ⁻¹B), X]

· · · [Eⁱ⁺¹B, X]

pf

_∂f

[EⁱB, M_∗f(EA, X)]

≈ Hf

[EⁱB,Map_f(EA, X)] · · ·, where ∂f is the boundary operator in the exact ho- motopy sequence for the evaluation fibration ω_f, the map H_f is the adjoint isomorphism (also called the Hurewicz isomorphism), and pf is the generalized Whitehead product with f, i.e., it holds p_f(g) = [f, g]

for anyg∈[Eⁱ⁺¹B, X].

Recall that Gottlieb groups G_n(X) for n ≥ 1 of a space X have been defined in Gottlieb^{7, 8)} as evaluation subgroups Gn(X) = Im(ev_∗ : π_n(Map(X, X),idX)→π_n(X)) of then-th homotopy groupπn(X). For a wide class of spacesX, the group Gn(X) is the subgroup of πn(X) containing all ele- mentsf :Sⁿ → X such that f ∨idX :Sⁿ∨X → X extends (up to homotopy) to a mapF :Sⁿ×X→X, i.e., the diagram

Sⁿ∨X

f∨idX

X

Sⁿ×X

F

commutes (up to homotopy). It is easy to observe that Gn(X) =πn(X) providedX is anH-space.

Further, Whitehead center groups P_n(X) for n ≥ 1 defined in Gottlieb⁸⁾ consist of all elements f ∈πn(X) such that the Whitehead product [f, g] = 0 for any g ∈π_m(X) andm ≥1. Certainly, G_n(X)⊆ Pn(X) for alln≥ 1. Notice that for any f ∈Pn(X) and i ≥ 1, Theorem 2.2 implies the short exact se-

quence

0→πn+i(X)→πi(Map_f(Sⁿ, X))→πi(X)→0.

(2.1) More generally, a map f : A → X such that f∨idX :A∨X →X extends (up to homotopy) to a mapF :A×X→X, i.e., such that the diagram

A∨X

f∨idX

X

A×X

F

commutes (up to homotopy) is called cyclic. In par- ticular, a map g : EA → EX is cyclic if and only if the generalized Whitehead product [g, ιEX] = 0, whereι_EXis the identity map on the spaceEX. Write G(A, X) for the set of all cyclic mapsA→X and no- tice thatG(A, X) is a subgroup of [A, X] provided A is a co-H-space.

Yoon observed a connection betweenG(EA, X) and path-components of Map(EA, X). In particular, Yoon²³⁾ Theorems 4.5 and 4.9 imply:

Proposition 2.3 For any f ∈ [EA, X] the following are equivalent:

(1) the evaluation fibrations (Map_f(EA, X), ωf, X) and (Map₀(EA, X), ω0, X) are fibre homotopy equivalent;

(2) the evaluation fibration (Map_f(EA, X), ωf, X) has a section;

(3) f∈G(EA, X).

Proposition 2.4 If f, g ∈ [EA, X] and f + g ∈ G(EA, X) (or f − g ∈ G(EA, X)) then the evaluation fibrations (Map_f(EA, X), ωf, X) and (Map_g(EA, X), ωg, X) are fibre homotopy equivalent.

In particular, the path-componentsMap_f(EA, X)and Map_g(EA, X), Map_f(EA, X)andMap_−f(EA, X)are homotopy equivalent.

BecauseGn(X)⊆Pn(X), by Proposition 2.3 the sequence (2.1) splits and we get an isomorphism

π_i(Map_f(Sⁿ, X))≈π_n+i(X)⊕π_i(X) (2.2)

providedf∈Gn(X) fori≥1.

Then Lupton–Smith¹⁴⁾Corollary 2.5 have shown that the surjection

[A, X]→ {Map_f(A, X);f ∈ A, X} given by [f]→Map_f(A, X) yields a surjection

[A, X]/G(A, X)→ {Map_f(A, X);f ∈ A, Y}/, (2.3) providedAis a co-H-space, whereis the homotopy equivalence relation. In particular, ifX is anH-space then all path-components of Map(A, X) are homotopy equivalent. Further, forA=Sⁿ there is also a surjec- tion

π_n(X)/Gn(X)→ {Map_f(Sⁿ, X);f ∈ Sⁿ, X}/. (2.4)

3. Path-components of the mapping space Map(S^m, M(A, n))

Given an abelian group A and n ≥ 2, write M(A, n) for the Moore space of type (A, n) and no- tice that EM(A, n) = M(A, n+ 1). If A = Z^k, the cyclic group of orderk, thenM(Zk, n) =Sⁿ∪kιneⁿ⁺¹. Write i_n :Sⁿ →M(Zk, n) for the canonical inclusion map andpn:M(Zk, n)→Sⁿ⁺¹ for the pinching map.

Proposition 3.1 (1) Map(S^m, M(A, m)) is path- connected.

Iff ∈π_m(M(A, n))then:

(2)there is an isomorphism

π_i(Map_f(S^m, M(A, n)))≈π_i+m(M(A, n)) fori < n−1and f∈πm(M(A, n));

(3) if the suspension map E : πi+m+k(M(A, n)) → π_i+m+k+1(M(A, n+ 1)) fork=−1,0 is a monomor- phism then there is an isomorphism

π_i(Map_f(S^m, M(A, n))/πi+m(M(A, n))≈

πi−1(M(A, n));

(4)there is an isomorphism πn−1(Map_f(S^m, M(A, n)))≈

π_m+n−1(M(A, n))/Im∂_f.

In particular, there is an isomorphism

π_n−1(Map_in(Sⁿ, M(Zk, n)))≈π_2n−1(M(Zk, n)) ifkandnare odd and

πn−1(Mapin(Sⁿ, M(Zk, n)))≈

π_2n−1(M(Zk, n))/

[in, i_n] ifkandnare others;

(5)there is the short exact sequence 0→π_m+n(M(A, n))/Im∂_f

→πn(Map_∗f(S^m, M(A, n)))→πn(M(A, n))→0;

(6) if π_i+1(M(A, n)) = 0 for some i ≥ 0 then there is a monomorphism πi+m(M(A, n)) → π_i(Mf(S^m, M(A, n)).

Proof. (1): Becauseπ_m(M(A, n)) = 0 form < n, we deduce that Map(S^m, M(A, m)) is path-connected.

Let now m ≥ n and given f ∈

Map(S^m, M(A, n)), consider the evaluation fibra- tion Map_f∗(S^m, M(A, n)) → Map_f(S^m, M(A, n)) → M(A, n). Then, Theorem 2.2 yields the commutative diagram

π_i+m(M(A, n))

· · · πi+1(M(A, n))

pf

∂f

πi(Map_∗f(S^m, M(A, n)))

Hf

πi(Map_f(S^m, M(A, n))) · · ·. (2): The equation π_i(M(A, n)) = 0 for i < n implies an isomorphism π_i(Map_f(S^m, M(A, n))) ≈ πi+m(M(A, n)) fori < n−1 andf ∈πm(M(A, n)).

(3): If the suspension maps E : π_i+m(M(A, n)) → πi+m+1(M(A, n+ 1)) and E : πi+m−1(M(A, n)) → πi+m(M(A, n + 1)) are monomorphisms then the maps p_f : π_i+1(M(A, n)) → π_i+m(M(A, n)) and pf : πi(M(A, n))→ πi+m−1(M(A, n)) are trivial and the result follows from the homotopy exact sequence above.

(4): The exact sequence

· · · →πn(M(A, n))^∂→^f πm+n−1(M(A, n))

→π_n−1(Map_f(S^m, M(A, n)))→0

yields an isomorphism πn−1(Map_∗f(S^m, M(A, n))) ≈ π_m+n−1(M(A, n))/Im∂_f.

(5): The exact sequence

· · · →π_n+1(M(A, n))→^∂f π_m+n(M(A, n))

→π_n(Map_f(S^m, M(A, n)))→π_n(M(A, n))→0 yields the short exact sequence

0→π_m+n(M(A, n))/Im∂_f

→πn((Map_f(S^m, M(A, n))→πn(M(A, n))→0.

(6): Ifπ_i+1(M(A, n)) = 0 for somei≥0 then by the homotopy sequence above there is a monomorphism πi+m(M(A, n))→πi(Mf(S^m, M(A, n))).

Certainly, πn(M(A, n)) = A for any abelian groupAand it is easy to see thatπn+1(M(Z2^k, n)) = Z2

i_n+1η_n

for k≥ 1. Using Golasi´nski and Mukai⁶⁾ Proposition 3.19, we can state:

Proposition 3.2 If the A is a finite group with odd order then π_n+1(M(A, n)) = 0forn≥3.

The stable homotopy groupsπm(M(Zk, n)) have been determined by Araki and Toda¹⁾and the follow- ing result in the stable range follows:

Proposition 3.3 (Araki and Toda¹⁾ (4.2)) (1)π_n(M(Zk, n)) =Zk

i_n

for allk;

(2)πn+1(M(Zk, n)) =





0, forkodd;

Z2

i_nη_n

, fork≡0,2 (mod 4).

Applying Golasi´nski and Mukai⁶⁾ Proposition 3.10, we can state:

Corollary 3.4 (1)Gn(M(Zk, n)) = 0forn≥2;

(2)G₃(M(Zk,2)) = 2π3(M(Zk,2))and Gn+1(M(Zk, n)) = 0forn≥3.

Then, in view of Proposition 3.1, we conclude:

Corollary 3.5 If A is a finite abelian group with an odd order and f ∈ πm(M(A, n)) then the canoni- cal map π_i+m(M(A, n)) → π_i(Mf(S^m, M(A, n)) is a monomorphism for n = 3 and an isomorphism for n≥4.

Next, Proposition 3.2 yields:

Corollary 3.6 If A is a finite group with odd order then the space Map(Sⁿ⁺¹, M(A, n)) is path-connected forn≥3.

IfAis a finite abelian group then the homotopy groups π_m(M(A, n)) for m ≥ 1 are finite and conse- quently, the cardinality of the set

{Map(S^m, M(A, n));f∈ S^m, M(A, n)/ is bounded above by the orderπ_m(M(A, n)). Further, by Smith¹⁷⁾there is a surjection

πm(M(A, n))/Gm(M(A, n))

→ {Map(S^m, M(A, n));f ∈ S^m, M(A, n)}/. Because by Golasi´nski and Mukai⁶⁾ Corollary 3.5, the order [in, ι_M(Zk,n)] = k we deduce that Gn(M(Z^k, n)) = 0.

Proposition 3.7 The cardinality of the set

S^n+l, M(Zk, n)/ is:

(1)bounded above by ⁿ₂ + 1 ifnis even and ⁿ⁺¹₂ ifn is odd forl= 0;

(2)one if kis odd and two for l= 1andkeven.

Proof. (1): Because π_n(M(Zk, n)) = Zk, Propo- sition 2.3 implies that the cardinality of the set Sⁿ, M(Zk, n))/ is bounded above by ⁿ₂ + 1 if nis even and ⁿ⁺¹₂ ifnis odd.

(2): The space Map(Sⁿ⁺¹, M(Zk, n)) is path-connected ifkis odd becauseπ_n+1(M(Zk, n)) = 0 ifkis odd.

Let now k be even. But π₃(M(Zk,2)) = Z4

i2η2

andπn+1(M(Zk, n)) =Z2 inηn

if kis even and n ≥ 3. By Golasi´nski and Mukai⁶⁾ Corollary 3.11 it holds G3(M(Z^k,2)) = Z²

2i2η2

and this im- plies that the path-components Map₀(S³, M(Zk,2)) and Map_2i2η2(S³, M(Zk,2)) are homotopy equiva- lent and by Proposition 2.3 the path-components Map_i2η2(S³, M(Zk,2)) and Map_3i2η2(S³, M(Zk,2)) are homotopy equivalent as well.

Given f ∈ π3(M(Zk,2)) consider the fibra- tion Map_∗f(S³, M(Zk,2)) → Map_f(S³, M(Zk,2) →

M(Zk,2).Because [i2η2,η˜2] is a generator of the group π₆(M³)≈(Z2)⁵, the exact sequences

0→π5(M(Zk,2))→π3(Map₀(S³,(M(Zk,2))

→π₃(M(Zk,2))→0 and

0→π5(M(Zk,2))/

[i2η2,η˜2]

→π₃(Map_i2η2(S³,(M(Zk,2))→π₃(M(Zk,2)) yield that the homotopy groups

π₃(Map₀(S³,(M(Zk,2)) and π₃(Map_i2η2(S³, (M(Z^k,2)) are not isomorphic. This implies that the path-components Map₀(S³,(M(Zk,2) and Map_i2η2(S³, (M(Zk,2) are not homotopy equivalent and the cardi- nality of the setS³, M(Zk,2)/is two.

Ifn≥3 then [inηn,η˜n]= 0, where ˜ηn:Sⁿ⁺²→ M(Zk, n) is the coextension ofη_n:Sⁿ⁺¹→Sⁿ. Given f ∈πn+1(M(Zk, n)) the fibration

Map_∗f(Sⁿ⁺¹, M(Zk, n))→Map_f(Sⁿ⁺¹, M(Zk, n)

→M(Zk, n)

yields the exact sequence

· · · →π_n+2(M(Zk, n))→π_2n+1(M(Zk, n))

→πn+1(Map_f(Sⁿ⁺¹, M(Zk, n)))

→πn+1(M(Zk, n))→ · · ·. This implies exact sequences:

0→π2n+1(M(Zk, n))→

π_n+1(Map₀(Sⁿ⁺¹, M(Zk, n)))→π_n+1(M(Zk, n))→0 and

0→π_2n+1(M(Zk, n))/

[inη_n,η˜_n]

→

π_n+1(Map_inηn(Sⁿ⁺¹, M(Zk, n)))→π_n+1(M(Zk, n)).

Because πn+1(M(Zk, n)) = Z2

inηn, we deduce that π_n+1(Map₀(Sⁿ⁺¹, M(Zk, n))) and πn+1(Map_inηn(Sⁿ⁺¹, M(Zk, n))) are not isomorphic.

Hence, the path-components Map₀(Sⁿ⁺¹, M(Zk, n)) and Map_inηn(Sⁿ⁺¹, M(Zk, n)) are not homotopy equivalent and the set Sⁿ⁺¹, M(Zk, n)/ contains exactly two elements and the proof is complete.

4. Path-components of the mapping space Map(M(A, m),Sⁿ)and Map(M(A, m), M(B, n))

Writing χ(X) for the Euler characteristic of the spaceX, we mimic Becker and Gottlieb³⁾Remarks and (8.8) Corollary to get:

Proposition 4.1 If X is a path-connected co-H- space, α∈G(X, Y)andχ(Y)<∞thenχ(Y)E^∞α= 0.

Proof. Ifα ∈G(X, Y) then there is a mapF :X× Y →Y such that the diagram

X∨Y

α∨ιY

Y

X×Y

F

commutes up to homotopy. Further, because X is path-connected, for any x ∈ X there is a path σ : I → X such that σ(0) =x0 and σ(1) =x. Then the mapG :I×Y →Y given byG(t, y) =F(σ(t), y) for (t, y)∈I×Y yields a homotopy between the identity map ιY and the restriction F|x×Y : Y → Y. Hence, F|s×Y is a homotopy equivalence for anyx∈X.

Write H(Y) for the space of maps homotopic to ιY andω:H(Y)→Y for the evaluation map at a base point ofY. Then the adjoint ofF :X×Y →Y yields a map ˜F : X → H(Y) and consequently we get the factorization

X ^α

F˜

Y

H(Y).

ω

Thus, by means of Becker and Gottlieb³⁾ (1.1) Theorem and Remarks, we derive thatχ(Y)E^∞α= 0 and the result follows.

Because χ(M(A, n)) = 1 provided A is a finite group andχ(S²ⁿ) = 2, we conclude thatE^∞α= 0 for α∈G(X, M(A, n)) and 2E^∞α= 0 forα∈G(X,S²ⁿ).

Ifnis odd then [2ιn, ιn] = 0 and this implies that [ιn,2f] = 0 for anyf ∈[EA,Sⁿ], that is, 2[EA,Sⁿ]⊆ G(EA,Sⁿ).

By means of Toda²⁰⁾Theorem 4.4, it holds:

ι_M(Zk,n)=





2k, for k≡2 (mod 4);

k, for k≡2 (mod 4).

This implies:

Remark 4.2 IfAis a pointed space then 2k[EA, M(Zk, n)]⊆G(EA, M(Zk, n)) ifk≡2 (mod 4)and

k[EA, M(Zk, n)]⊆G(EA, M(Zk, n)) ifk≡2 (mod 4).

In particular,

2kπm(M(Zk, n))⊆G_m(M(Zk, n)) ifk≡2 ( mod 4 )and

kπ_m(M(Zk, n))⊆G_m(M(Zk, n)) ifk≡2 (mod 4)form≥1.

Then, by Equation (2.3), there is a surjection [EA, M(Zk, n)]/2k[EA, M(Zk, n)]

→ {Map_f(EA, M(Zk, n));f ∈ EA, M(Zk, n)}/ ifk≡2 (mod 4) and

[EA, M(Zk, n)]/k[EA, M(Zk, n)]

→ {Map_f(EA, M(Zk, n));f ∈ EA, M(Zk, n)}/ ifk≡2 (mod 4).

Now, write Mⁿ = M(Z2, n − 1) and no- tice that Eⁿ⁻²RP² = M(Z2, n − 1) for n ≥ 3, where RP² denotes the real projective plane. Re- call that Mukai¹⁶⁾ Theorems 3.1–3.3 describe the sta- ble groups colimnπ_n+k(Mⁿ), colimn[M^n+l,Sⁿ] and colimn[M^n+l, Mⁿ]. The stable homotopy classes colimn[M(Zk, n + l),Sⁿ], and colimn[M(Zk, n + l), M(Zk, n)] have been determined by Araki and Toda¹⁾ and they obtained the following results.

Proposition 4.3 (Araki and Toda¹⁾ (4.2)) (1) [M(Zk, n−1),Sⁿ] =Zk

p_n−1

for allk;

(2) [M(Zk, n),Sⁿ] =Z2 ηnpn

for allk.

Proposition 4.4 (Araki and Toda¹⁾ Theorem 4.1) (1) [M(Z^k, n−1), M(Z^k, n] =Z^k

pn−1in

for all q;

(2) [M (Zk, n), M(Zk, n)] =







 Zk

ι_M(Zk,n)

, forkodd;

Z2k

ι_M(Zk,n)

, fork≡2 (mod 4);

Z2 inηnpn

⊕Zk

ι_M(Zk,n)

, fork≡0 (mod 4).

2010 Mathematical Subject Classification. Primary 46T10, 54C35; secondary 58D15, 55P99

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