The group of self-homotopy equivalences of the m-fold smash product of a space

Let E(X) be the set of homotopy classes of self-homotopy equivalences of a space X. The set E(X) is a group by composition of homotopy classes. We study the group E(X∧m) for the m-fold smash product X∧m. We show that the two obvious homomorphisms φ:Sm→E(X∧m) and ψ:E(X)m→E(X∧m) define a homomorphism Ψ:E(X)m⋊Sm→E(X∧m) for any space X, where E(X)m⋊Sm is the semi-direct product of the product group E(X)m by the symmetric group Sm. We show that in most cases the homomorphism φ:Sm→E(X∧m) is a monomorphism and the kernel of Ψ is isomorphic to the kernel of ψ. The injectivity of Ψ is established for the complex projective n-space CPn (n≥2), and hence, E((CPn)∧m) contains a subgroup isomorphic to {±1}m⋊Sm. Sufficient conditions for Ψ to be injective are obtained for the Eilenberg-MacLane complex K(Ar,n) where A is a subring of Q or a ring Z/k (k≥2) and Ar is the free A-module of rank r. From this result, we see that E(K(Ar,n)∧m) contains a subgroup isomorphic to GLr(A)m⋊Sm in many cases.