On the group

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 π i ( X ) for i ⩽ N . We describe a new kind of factorization of Aut ♯ N ( X 1 × ⋯ × X n ) into a product of simpler subgroups.