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Certain Homotopy Properties Related to $\Text {Map}(\Sigma^ N\Mathbb {C CERTAIN HOMOTOPY PROPERTIES RELATED TO map(ΣnCP 2,Sm) Jin-ho Lee Abstract. We compute cohomotopy groups of suspended complex plane πn+m(ΣnCP 2) for m = 6, 7, 8. Using these results, we classify path components of the spaces map(ΣnCP 2,Sm) up to homotopy equivalent. m We determine the generalized Gottlieb groups Gn(CP 2,S ). Finally, we compute homotopy groups of mapping spaces map(ΣnCP 2,Sm; f) for all generators [f] of [ΣnCP 2,Sm] and Gottlieb groups of mapping components containing constant map map(ΣnCP 2,Sm; ∗). 1. Introduction Let X and Y be based topological spaces. A major object of homotopy theory is to study [X, Y ], the set of homotopy equivalence classes of based maps. In general, if Y is a co-H-group, [X, Y ] has group structure. Let ΣX be the suspension of X. Since every suspended space ΣX is co-group, [ΣX, Y ] has group structure. If ΣX is a sphere Sn,[Sn, Y ] is the n-th homotopy group of Y . On the other hand, [X,Sn] is called the n-th cohomotopy set of X and denoted by πn(X). Moreover, if X is a co-H-group, the cohomotopy set is a group and called n-th cohomotopy group of X. Homotopy groups and cohomotopy groups have been studied by many authors and are a major object in algebraic topology. Another major object of homotopy theory is to investigate the set of (un- based) maps f : X → Y . We denote map(X, Y ) to be the set of all con- tinuous maps from X to Y equipped with compact-open topology. Then we write map(X, Y ; f) for the path-component of map(X, Y ) containing a map f : X → Y . Important cases are map(X, Y ; ∗) and map(X,X;1) where ∗ : X → Y is the constant map and 1 : X → X is the identity map on X. D. Harris proved that every topological space X is the space of path com- ponents of a stratifiable space S(X) [9]. Thus it is important to study path components of mapping spaces to analyze topological spaces. Lang proved that if X is a suspended space, then all path components of based mapping space map∗(X, Y ) have the same homotopy type [15, Theo- arXiv:1501.03242v2 [math.AT] 6 Oct 2015 rem 2.1]. Whitehead proved that map(Sn,Sm; f) is homotopy equivalent to 2010 Mathematics Subject Classification. Primary 54C35,55P15,55Q52,55Q55. Key words and phrases. Complex plane, Cohomotopy group, Toda Bracket, mapping space, path-component. 1 2 JIN-HO LEE n m n m map(S ,S ; ∗) if and only if wf has a section, where ∗ : S → S is a constant map [30, Theorem 2.8]. Smith proved a rational homotopy equivalence map(X,X; ∗) ≃Q X × map∗(X,X : ∗) for a “nice” space X [26]. Lupton and Smith proved that map(X, Y ; f) ≃ map(X, Y ; f + d) for a CW co-H-space X and any CW complex Y , where d : X → Y is a cyclic map [16, Theorem 3.10]. Recently, Gatsinzi [3] proved that the dimension of the rational Gottlieb group of the universal cover map(X,S2n; f) of the function space map(X,S2n; f) is at least equal to the dimensiong of H˜ ∗(X; Q) under several assumptions. Lupton and Smith [16] showed that Gn(map∗(X, Y ; ∗)) =∼ Gn(Y ) ⊕ Gn(X, Y ). Maruyama and Oshima [18] determined homotopy groups of map∗(SU(3),SU(3); ∗), map∗(Sp(2),Sp(2); ∗) and map∗(G2, G2; ∗). In Section 2, we present some basic knowledge of composition methods [28]. We review a mapping cone sequence and a Puppe sequence related to the sus- pended complex projective plane and discuss the concept of cyclic maps and its properties. Also we recall the Toda bracket and its properties. In Sections 3, 4 and 5, we compute πn(Σn+kCP 2) for k =6, 7. As a result, we obtain the following results: case n 2 3 4 5 6 7 πn(Σn+6CP 2) 2+15 2+3 8+2+32 +5 4+9 42 +9+3 4+3 case n 8 9 10 11 n ≥ 12 πn(Σn+6CP 2) 42 +32 4+3 2+3 2+3 2 case n 2 3 4 5 πn(Σn+7CP 2) 2 + 3 22 + 21 4 + 23 + 21 4 + 22 + 63 case n 6 7 8 9 πn(Σn+7CP 2) 4 + 22 + 63 8 + 22 8 + 22 8 + 22 case n 10 11 12 n ≥ 13 πn(Σn+7CP 2) ∞ +8+2+63 8 + 2 + 63 ∞ +8+2+63 8 + 2 + 63 case n 2 3 4 5 6 7 8 πn(Σn+8CP 2) 22 + 21 2 + 3 8 + 2 4 + 9 8+4+9+3 8 + 3 82 + 22 case n 9 10 11 12 13 n ≥ 14 πn(Σn+8CP 2) 8 + 3 8 + 2 22 2 + 3 2+3 3 n m CERTAIN HOMOTOPY PROPERTIES RELATED TO map(Σ CP 2,S ) 3 k where an integer n denotes the cyclic group Zn, (s) denotes the k-times di- rect sum of Zs, ∞ denotes the group of integers Z and “+” denotes the direct sum of abelian groups. In section 6, using our result in section 3, 4 and 5, we determine nth- 2 2 homotopy groups of function spaces map∗(CP , CP ; ∗) for 4 ≤ n ≤ 13. In Section 7, we apply our results to the classification of path components of mapping spaces up to homotopy equivalent and evaluation fibrations up to fibre homotopy equivalent. Hansen proved that the evaluation fibration wf : map(ΣX, ΣY ; f) → ΣY has a section if and only if [f,idΣY ] = 0, where [ , ] is the generalized Whitehead product [8]. Lupton and Smith proved that the following statements are all equivalent: (1) a map f : X → Y is cyclic, (2) wf has a section, and (3) two fibrations wf and w∗ are fiber homotopy equivalent, where ∗ is a constant map [16]. Moreover, we apply our results to the formulation of generalized Gottlieb groups from suspended complex plane to sphere and Gottlieb groups of path components of constant map. We use the notation of [28, 11] freely. 2. Preliminaries 2 2 4 The complex projective plane CP is defined by the mapping cone S ∪η2 e 3 2 k−2 where η2 : S → S is the Hopf fibration. We denote ηk = Σ η2 for k ≥ 2. Consider a Puppe sequence η2 i p η3 i S3 −→ S2 −→ CP 2 −→ S4 −→ S3 −→·Σ · · , where i : S2 → CP 2 is the inclusion map and p : CP 2 → S4 is the collapsing map of S2 to a point. The Puppe sequence induces a long exact sequence of homotopy sets ∗ η n ∗ m n+3 m Σ p n 2 m ···→ πn+3(S ) −−−→ πn+4(S ) −−−→ [Σ CP ,S ] n ∗ ∗ Σ i m ηn+2 m −−−→ πn+2(S ) −−−→ πn+3(S ) → · · · . Then we have the short exact sequence n ∗ n ∗ ∗ Σ p n 2 m Σ i ∗ (1.1) 0 → Cokerηn+3 −−−→ [Σ CP ,S ] −−−→ Kerηn+2 → 0. 7 2 6 1 Let g6(C):Σ CP → S be the S -transfer map [23]. This map is the adjoint of the composite of inclusions .ΣCP 2 ֒→ SU(3) ֒→ SO(6) ֒→ Ω6S6 n 7 We set gn+6(C)=Σ g6(C) for n ≥ 1. Note that g6(C) = [ι6, ι6] ◦ Σ p + ν6 ◦ 2ι9 and 2gn(C)= νn ◦ 2ιn+3 for n ≥ 7 [11, Proposition 3.3]. 4 JIN-HO LEE Let G be an abelian group. For a prime p, we denote the p-primary parts of G by G(p). For an odd prime p, we have an isomorphism n 2 k ∼ k k (1.2) [Σ CP ,S ](p) = πn+2(S )(p) ⊕ πn+4(S )(p), n since πn+1(S ) is of order 2 for n ≥ 3 [28, Proposition 5.1]. The space ΣCP n−1 has reduced homology n−1 H∗(ΣCP )= Zp[x3, x5, ··· , x2n−1]. e n−1 There is a Steenrod operation on H∗(ΣCP ) given by e j P (x2r+1)= r!/(j! · (r − j)!) · x2r+jp+1 (2 ≤ r ≤ n − 1). It has been shown that there exists a wedge decomposition [22] p−1 n−1 ΣCP ≃ _ Ai(n) i=1 where Ai(n) has a reduced homology n H∗(Ai(n)) = Z/pZ{x2i+1, x2i+q+1, ··· , x2i+(rn−1)q+1 | ri = ⌊(n−i−1)/(p−1)⌋+1} e i n−1 that inherits a Steenrod operation from H∗(ΣCP ). e Consider elements α ∈ [Y,Z], β ∈ [X, Y ], and γ ∈ [W, X] which satisfy α ◦ β = 0 and β ◦ γ = 0. Let Cβ be the mapping cone of β, i : Y → Cβ and p : Cγ → ΣX be the inclusion and the shrinking map, respectively. We ∗ denote an extension α ∈ [Cβ,Z] of α satisfying i (α) = α and a coextension γ ∈ [ΣW, Cβ] of γ satisfying p∗(γ)=Σγ [28]. From the definition of extension, ane extension α exists if and onlye if α ◦ β = 0. Similarly, a coextension γ exists if and only if β ◦ γ =0. e We recall a useful relation between extensions and Toda brackets [28, Propo- sition 1.9]: Proposition 1. Let α ∈ [Y,Z], β ∈ [X, Y ], and γ ∈ [W, X] be elements which satisfy α ◦ β = 0 and β ◦ γ = 0. Let {α,β,γ} be the Toda bracket and p : Cγ → ΣW be the shrinking map. Then, we have α ◦ β ∈{α,β,γ}◦ p.
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