The group of self-homotopy equivalences of the m-fold smash product of a space

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• 作者：Hiroshi Kihara^a ; ^{kihara@u-aizu.ac.jp} ; Ken-ichi Maruyama^b ; ^{maruyama@faculty.chiba-u.jp} ; Nobuyuki Oda^c ; ¹ ; ^{odanobu@cis.fukuoka-u.ac.jp}
• 关键词：55P10
• 刊名：Topology and its Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：217
• 期：Complete
• 页码：70-80
• 全文大小：323 K
• 卷排序：217

文摘

Let E(X)E(X) be the set of homotopy classes of self-homotopy equivalences of a space X  . The set E(X)E(X) is a group by composition of homotopy classes. We study the group E(X∧m)E(X∧m) for the m  -fold smash product X∧mX∧m. We show that the two obvious homomorphisms φ:Sm→E(X∧m)φ:Sm→E(X∧m) and ψ:E(X)m→E(X∧m)ψ:E(X)m→E(X∧m) define a homomorphism Ψ:E(X)m⋊Sm→E(X∧m)Ψ:E(X)m⋊Sm→E(X∧m) for any space X  , where E(X)m⋊SmE(X)m⋊Sm is the semi-direct product of the product group E(X)mE(X)m by the symmetric group SmSm. We show that in most cases the homomorphism φ:Sm→E(X∧m)φ:Sm→E(X∧m) is a monomorphism and the kernel of Ψ is isomorphic to the kernel of ψ. The injectivity of Ψ is established for the complex projective n  -space CPnCPn(n≥2)(n≥2), and hence, E((CPn)∧m)E((CPn)∧m) contains a subgroup isomorphic to {±1}m⋊Sm{±1}m⋊Sm. Sufficient conditions for Ψ to be injective are obtained for the Eilenberg–MacLane complex K(Ar,n)K(Ar,n) where A   is a subring of QQ or a ring Z/kZ/k(k≥2)(k≥2) and ArAr is the free A-module of rank r  . From this result, we see that E(K(Ar,n)∧m)E(K(Ar,n)∧m) contains a subgroup isomorphic to GLr(A)m⋊SmGLr(A)m⋊Sm in many cases.