ON FACTORIZATIONS OF THE SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES

For a pointed space X, the subgroups of self-homotopy equivalences Aut]N (X), AutΩ(X), Aut∗(X) and AutΣ(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 ≤ ∞, AutΣ(X) is the group of all the above g such that Σg = id. We will prove that AutΩ(X1×· · ·×Xn) has two factorizations similar to those of Aut]N (X1×· · ·×Xn) in reference [10], and that AutΣ(X1∨· · ·∨Xn), Aut∗(X1∨· · ·∨Xn) also have factorizations being dual to the former two cases respectively.