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 ⁎ ∈ Im Ψ . For the R -localized X R of an ( n + 1 ) -connected and ( 3 n + 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 X R and we show that f ⁎ ∈ Im Φ if and only if f ⁎ is a strong automorphism. Consequently we prove that E ( X R ) ( E ⁎ ( X ) ) R ≅ B ( X R ) , where E ⁎ ( X ) is the subgroup of the elements that induce the identity on homology and B ( X R ) is the subgroup of the strong automorphisms. [ABSTRACT FROM AUTHOR]