On the group of self-homotopy equivalences of (n + 1)-connected and (3n + 2)-dimensional CW-complex

Let X be a CW complex, E(X) the group of homotopy classes of homotopy self-equivalences of X and Ψ:E(X)→aut(H⁎(X,Z)) the map sending [α] to H⁎(α). This paper deals with the following question: Characterize f⁎∈aut(H⁎(X,Z)) such that f⁎∈ImΨ. For the R-localized XR of an (n+1)-connected and (3n+2)-dimensional CW-complex X; n≥2, where R is a certain subring of Q we define the notion of strong automorphism in aut(H⁎(X,Z)), in term of the Whitehead exact sequence of the Anick model of XR and we show that f⁎∈ImΦ if and only if f⁎ is a strong automorphism. Consequently we prove that E(XR)(E⁎(X))R≅B(XR), where E⁎(X) is the subgroup of the elements that induce the identity on homology and B(XR) is the subgroup of the strong automorphisms.