On the spaces of self homotopy equivalences of certain CW complexes (Q1056954)

For any CW complex Z, let G(Z) (resp. \(G_ 0(Z))\) denote the space of self-homotopy equivalences of Z (resp. preserving base point). The author notes that G(X\(\times Y)\) and \(G_ 0(X\times Y)\) each have the weak homotopy type of a product of 4 spaces in the case where Y is n-connected \((n>1)\) and dim \(X\leq n\) or \(\pi_ i(X)=0\) for \(i>n\). For example: \(G_ 0(X\times Y)\sim G_ 0(X)\times G_ 0(Y)\times (G(X),1)^{(Y,*)}\times (G(Y),1)^{(X,*)}\) and in particular \(G_ 0(K(\pi,n)\times Y)\sim aut \pi \times G_ 0(Y)\times (G(Y),1)^{(K(\pi,n),*)}\) and \(\epsilon\) (K(p,n)\(\times Y)\) is a semi-direct product (aut \(\pi\) \(\times \epsilon (Y)){\tilde \times}[K(\pi,n),G(Y)]\). Some related results are given.