THE GROUP OF SELF-HOMOTOPY EQUIVALENCES OF A SIMPLY CONNECTED an‎d 4-DIMENSIONAL CW-COMPLEX

Let X be a CW complex, E(X) the group of homotopy classes of self-homotopy equivalences of X and E∗(X) its subgroup of the elements that induce the identity on homology. This paper deals with the problem 19 in [Contemp. Math., 519 (2010), 217-230]. Given a group G, find a space X such that E(X) E∗(X) = G. For a simply connected and 4-dimensional CW-complex X we define a group B 4 ⊂ aut(H∗(X,Z)) in term of the Whitehead exact sequence of X and we show that this problem has a solution if G ∼= B 4 for some space X.