On factorizations of the subgroups of self-homotopy equivalences
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For a pointed space X, the subgroups of self-homotopy equivalences Aut#N (X), AutΩ(X), Aut*(X) and Aut∑(X) are considered, where Aut#N (X) is the group of all self-homotopy classes f of X such that f# = id : πi(X) → πi(X) for all i ≤ N ≤ ∞, AutΩ(X) is the group of all the above f such that Ωf = id; Aut*(X) is the group of all selfhomotopy classes g of X such that g* = id : Hi(X) → Hi(X) for all i ≤ ∞, Aut∑(X) is the group of all the above g such that ∑g = id. We will prove that AutΩ(X₁×...×Xn) has two factorizations similar to those of Aut#N (X₁×...×Xn) in reference [10], and that Aut§(X₁∨...∨Xn), Aut*(X₁∨...∨ Xn) also have factorizations being dual to the former two cases respectively.
Abstract
For a pointed space X, the subgroups of self-homotopy equivalences Aut#N (X), AutΩ(X), Aut*(X) and Aut∑(X) are considered, where Aut#N (X) is the group of all self-homotopy classes f of X such that f# = id : πi(X) → πi(X) for all i ≤ N ≤ ∞, AutΩ(X) is the group of all the above f such that Ωf = id; Aut*(X) is the group of all selfhomotopy classes g of X such that g* = id : Hi(X) → Hi(X) for all i ≤ ∞, Aut∑(X) is the group of all the above g such that ∑g = id. We will prove that AutΩ(X₁×...×Xn) has two factorizations similar to those of Aut#N (X₁×...×Xn) in reference [10], and that Aut§(X₁∨...∨Xn), Aut*(X₁∨...∨ Xn) also have factorizations being dual to the former two cases respectively.
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