On the group Aut♯(X1×⋯×Xn)

Let N⩾dim(X) and let Aut♯N(X) denote the group of homotopy classes of self-maps of X, which induce identity automorphisms of the homotopy groups πi(X) for i⩽N. We describe a new kind of factorization of Aut♯N(X1×⋯×Xn) into a product of simpler subgroups.